Walsh matrix

From Rosetta Code
Walsh matrix is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
This page uses content from Wikipedia. The original article was at Walsh matrix. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)


A Walsh matrix is a specific square matrix of dimensions 2n, where n is some particular natural number. The elements of the matrix are either +1 or −1 and its rows as well as columns are orthogonal, i.e. dot product is zero. Each row of a Walsh matrix corresponds to a Walsh function.

Walsh matrices are a special case of Hadamard matrices. The naturally ordered Hadamard (Walsh) matrix is defined by the recursive formula below, and the sequency-ordered Hadamard (Walsh) matrix is formed by rearranging the rows so that the number of sign changes in a row is in increasing order.


To generate a naturally ordered Walsh matrix

Matrices of dimension 2k for k ∈ N are given by the recursive formula:

and in general

for 2 ≤ k ∈ N, where ⊗ denotes the Kronecker product.


Task
  • Write a routine that, given a natural number k, returns a naturally ordered Walsh matrix of order 2k.
  • Display a few sample generated matrices.
Traditionally, Walsh matrices use 1 & -1 to denote the different cell values in text mode, or green and red blocks in image mode. You may use whichever display mode is most convenient for your particular language.


Stretch
  • Also, optionally generate sequency ordered Walsh matrices.
A sequency ordered Walsh matrix has the rows sorted by number of sign changes.


See also


ALGOL 68

BEGIN # construct Walsh Matrices                                             #
  CO BEGIN code from the Kronecker product task                             CO
    # multiplies in-place the elements of the matrix a by the scaler b       #
    OP   *:= = ( REF[,]INT a, INT b )REF[,]INT:
    BEGIN
        FOR i FROM 1 LWB a TO 1 UPB a DO
            FOR j FROM 2 LWB a TO 2 UPB a DO
                a[ i, j ] *:= b
            OD
        OD;
        a
    END # *:= # ;
    # returns the Kronecker Product of the two matrices a and b              #
    # the result will have lower bounds of 1                                 #
    PRIO X = 6;
    OP   X = ( [,]INT a, b )[,]INT:
    BEGIN
        # normalise the matrices to have lower bounds of 1                   #
        [,]INT m = a[ AT 1, AT 1 ];
        [,]INT n = b[ AT 1, AT 1 ];
        # construct the result #
        INT r 1 size = 1 UPB n;
        INT r 2 size = 2 UPB n;
        [ 1 : 1 UPB m * 1 UPB n, 1 : 2 UPB m * 2 UPB n ]INT k;
        FOR i FROM 1 LWB m TO 1 UPB m DO
            FOR j FROM 2 LWB m TO 2 UPB m DO
                ( k[ 1 + ( ( i - 1 ) * r 1 size ) : i * r 1 size
                   , 1 + ( ( j - 1 ) * r 2 size ) : j * r 2 size
                   ] := n
                ) *:= m[ i, j ]
            OD
        OD;
        k
    END # X # ;
  CO END   code from the Kronecker product task                             CO
    # returns a Walsh matrix of oreder n                                     #
    OP   WALSH = ( INT n )[,]INT:
         BEGIN
            [,]INT w1 = ( (  1,  1 )
                        , (  1, -1 )
                        );
            FLEX[ 1 : 0, 1 : 0 ]INT result := 1;
            FOR order TO n DO
                result := result X w1
            OD;
            result
         END # WALSH # ;
    # returns Walsh matrix a sorted into sequency order                     #
    OP   SEQUENCYSORT = ( [,]INT a )[,]INT:
         BEGIN
            # sort the rows of the matrix into order of the number of sign  #
            # changes in the row                                            #
            [,]INT w = a[ AT 1, AT 1 ]; # normalise the matrix to have      #
                                        # lower bounds of 1                 #
            [ 1 : 1 UPB w, 1 : 2 UPB w ]INT result;
            # construct the resullt with the rows in order of the number of #
            # the number of sign changes in the original                    #
            # note the number of changes is unique and in 0 .. UPB a - 1    #
            FOR row FROM 1 TO 1 UPB w DO
                INT changes := 0;
                INT curr    := w[ row, 1 ];
                FOR col FROM 2 TO 2 UPB w DO
                    IF curr /= w[ row, col ] THEN
                        changes +:= 1;
                        curr     := w[ row, col ]
                    FI
                OD;
                result[ changes + 1, : ] := w[ row, : ]
            OD;
            result
         END # SEQUENCYSORT # ;
    CO returns r encoded with 1 = "_" and -1 = "#"                          CO
    OP   TOWSTRING = ( []INT r )STRING:
         BEGIN
            STRING result := "";
            FOR j FROM LWB r TO UPB r DO
                result +:= IF r[ j ] > 0 THEN "_" ELSE "#" FI
            OD;
            result
         END # TOWSTRING # ;
    # show the natural order and sequency order Walsh matrices of order 5    #
    [,]INT w5 = WALSH 5;
    [,]INT s5 = SEQUENCYSORT w5;
    FOR row FROM 1 TO 1 UPB s5 DO
        print( ( TOWSTRING w5[ row, : ], "        ", TOWSTRING s5[ row, : ], newline ) )
    OD
END
Output:
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C++

#include <algorithm>
#include <cstdint>
#include <iomanip>
#include <iostream>
#include <vector>

void display(const std::vector<std::vector<int32_t>>& matrix) {
	for ( const std::vector<int32_t>& row : matrix ) {
		for ( const int32_t& element : row ) {
			std::cout << std::setw(3) << element;
		}
		std::cout << std::endl;;
	}
	std::cout << std::endl;;
}

uint32_t sign_change_count(const std::vector<int32_t>& row) {
	uint32_t sign_changes = 0;
	for ( uint64_t i = 1; i < row.size(); ++i ) {
		if ( row[i - 1] == -row[i] ) {
			sign_changes++;
		}
	}
	return sign_changes;
}

std::vector<std::vector<int32_t>> walsh_matrix(const uint32_t& size) {
	std::vector<std::vector<int32_t>> walsh = { size, std::vector<int32_t>(size, 0) };
	walsh[0][0] = 1;

	uint32_t k = 1;
	while ( k < size ) {
		for ( uint32_t i = 0; i < k; ++i ) {
			for ( uint32_t j = 0; j < k; ++j ) {
				walsh[i + k][j] = walsh[i][j];
				walsh[i][j + k] = walsh[i][j];
				walsh[i + k][j + k] = -walsh[i][j];
			}
		}
		k += k;
	}
	return walsh;
}

int main() {
	for ( const std::string type : { "Natural", "Sequency" } ) {
		for ( const uint32_t order : { 2, 4, 5 } ) {
			uint32_t size = 1 << order;
			std::cout << "Walsh matrix of order " << order << ", " << type << " order:" << std::endl;
			std::vector<std::vector<int32_t>> walsh = walsh_matrix(size);
			if ( type == "Sequency" ) {
				std::sort(walsh.begin(), walsh.end(),
					[](const std::vector<int32_t> &row1, const std::vector<int32_t> &row2) {
						return sign_change_count(row1) < sign_change_count(row2);
					});
			}
			display(walsh);
		}
	}
}
Output:
Walsh matrix of order 2, Natural order:
  1  1  1  1
  1 -1  1 -1
  1  1 -1 -1
  1 -1 -1  1

Walsh matrix of order 4, Natural order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1

Walsh matrix of order 5, Natural order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1

Walsh matrix of order 2, Sequency order:
  1  1  1  1
  1  1 -1 -1
  1 -1 -1  1
  1 -1  1 -1

Walsh matrix of order 4, Sequency order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1

Walsh matrix of order 5, Sequency order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1

F#

// Walsh matrix. Nigel Galloway: August 31st., 2023
open MathNet.Numerics
open MathNet.Numerics.LinearAlgebra
let walsh()=let w2=matrix [[1.0;1.0];[1.0;-1.0]] in Seq.unfold(fun n->Some(n,w2.KroneckerProduct n)) w2
walsh() |> Seq.take 5 |> Seq.iter(fun n->printfn "%s" (n.ToMatrixString()))
Output:
1   1
1  -1

1   1   1   1
1  -1   1  -1
1   1  -1  -1
1  -1  -1   1

1   1   1   1   1   1   1   1
1  -1   1  -1   1  -1   1  -1
1   1  -1  -1   1   1  -1  -1
1  -1  -1   1   1  -1  -1   1
1   1   1   1  -1  -1  -1  -1
1  -1   1  -1  -1   1  -1   1
1   1  -1  -1  -1  -1   1   1
1  -1  -1   1  -1   1   1  -1

 1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
 1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1
 1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1
 1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1
 1   1   1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1
 1  -1   1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1
 1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1   1   1
 1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1   1  -1
..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..
 1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1   1   1   1   1
 1  -1   1  -1  -1   1  -1   1  -1   1  -1   1   1  -1   1  -1
 1   1  -1  -1  -1  -1   1   1  -1  -1   1   1   1   1  -1  -1
 1  -1  -1   1  -1   1   1  -1  -1   1   1  -1   1  -1  -1   1

 1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1  ..   1   1
 1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  ..   1  -1
 1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1  ..  -1  -1
 1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  ..  -1   1
 1   1   1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1   1  ..  -1  -1
 1  -1   1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1   1  ..  -1   1
 1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1   1   1   1  ..   1   1
 1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1   1  -1   1  ..   1  -1
..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..  ..
 1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1   1   1   1   1  -1  ..  -1  -1
 1  -1   1  -1  -1   1  -1   1  -1   1  -1   1   1  -1   1  -1  -1  ..  -1   1
 1   1  -1  -1  -1  -1   1   1  -1  -1   1   1   1   1  -1  -1  -1  ..   1   1
 1  -1  -1   1  -1   1   1  -1  -1   1   1  -1   1  -1  -1   1  -1  ..   1  -1

Factor

Works with: Factor version 0.99
USING: accessors formatting images.processing images.testing
images.viewer kernel math math.matrices math.matrices.extras
sequences sequences.extras sorting.extras ui ui.gadgets
ui.gadgets.borders ui.gadgets.labeled ui.gadgets.packs ;
IN: walsh

CONSTANT: walsh1 { { 1 1 } { 1 -1 } }
CONSTANT: red B{ 0 255 0 }
CONSTANT: green B{ 255 0 0 }

: walsh ( n -- seq )
    1 - walsh1 tuck '[ _ kronecker-product ] times ;

: sequency ( n -- seq )
    walsh [ dup rest-slice [ = not ] 2count ] map-sort ;

: seq>bmp ( seq -- newseq )
    concat [ 1 = red green ? ] B{ } map-concat-as ;

: seq>img ( seq -- image )
    dup dimension <rgb-image> swap >>dim swap seq>bmp >>bitmap ;

: <img> ( seq -- gadget )
    dup length 256 swap / matrix-zoom seq>img <image-gadget> ;

: info ( seq -- str )
    length dup log2 swap dup "Order %d  (%d x %d)" sprintf ;

: <info-img> ( seq -- gadget )
    [ <img> ] [ info ] bi <labeled-gadget> ;

: <pile-by> ( seq quot -- gadget )
    <pile> -rot [ <info-img> add-gadget ] compose each ; inline

: <natural> ( -- gadget )
    { 2 4 5 } [ walsh ] <pile-by> "Natural ordering"
    <labeled-gadget> ;

: <sequency> ( -- gadget )
    { 2 4 5 } [ sequency ] <pile-by> "Sequency ordering"
    <labeled-gadget> ;

: <walsh> ( -- gadget )
    <shelf> <natural> { 3 0 } <border> add-gadget
    <sequency> { 3 0 } <border> add-gadget ;

MAIN: [ <walsh> "Walsh matrices" open-window ]
Output:

J

kp1=: [: ,./^:2 */   NB. Victor Cerovski, 2010-02-26
walsh=: {{(_1^3=i.2 2)&kp1^:y 1}}
sequencyorder=: /: 2 ~:/\"1 ]

Small examples (time is not an issue here, but page estate is an issue): 

   walsh 0
1
   walsh 1
1  1
1 _1
   walsh 2
1  1  1  1
1 _1  1 _1
1  1 _1 _1
1 _1 _1  1
   walsh 3
1  1  1  1  1  1  1  1
1 _1  1 _1  1 _1  1 _1
1  1 _1 _1  1  1 _1 _1
1 _1 _1  1  1 _1 _1  1
1  1  1  1 _1 _1 _1 _1
1 _1  1 _1 _1  1 _1  1
1  1 _1 _1 _1 _1  1  1
1 _1 _1  1 _1  1  1 _1
   sequencyorder walsh 3
1  1  1  1  1  1  1  1
1  1  1  1 _1 _1 _1 _1
1  1 _1 _1 _1 _1  1  1
1  1 _1 _1  1  1 _1 _1
1 _1 _1  1  1 _1 _1  1
1 _1 _1  1 _1  1  1 _1
1 _1  1 _1 _1  1 _1  1
1 _1  1 _1  1 _1  1 _1

Java

import java.util.ArrayList;
import java.util.Collections;
import java.util.Comparator;
import java.util.List;
import java.util.stream.Collectors;
import java.util.stream.IntStream;

public final class WalshMatrix {

	public static void main(String[] args) {		
		for ( String type : List.of( "Natural", "Sequency" ) ) {	
			for ( int order : List.of( 2, 4, 5 ) ) {
			    int size = 1 << order;
			    System.out.println("Walsh matrix of order " + order + ", " + type + " order:");
			    List<List<Integer>> walsh = walshMatrix(size);
			    if ( type.equals("Sequency") ) {
			    	Collections.sort(walsh, rowComparator);
			    }
			    display(walsh);
			}
		}
	}
	
	private static List<List<Integer>> walshMatrix(int size) {
		List<List<Integer>> walsh = IntStream.range(0, size).boxed()
            .map( i -> new ArrayList<Integer>(Collections.nCopies(size, 0)) ).collect(Collectors.toList());		
		walsh.get(0).set(0, 1);
		
		int k = 1;
		while ( k < size ) {
	        for ( int i = 0; i < k; i++ ) {
	            for ( int j = 0; j < k; j++ ) {
	            	walsh.get(i + k).set(j, walsh.get(i).get(j));
	            	walsh.get(i).set(j + k, walsh.get(i).get(j));
	            	walsh.get(i + k).set(j + k, -walsh.get(i).get(j));
	            }
	        }
	        k += k;
		}
		return walsh;		
	}
	
	private static int signChangeCount(List<Integer> row) {
	    int signChanges = 0;
	    for ( int i = 1; i < row.size(); i++ ) {
	        if ( row.get(i - 1) == -row.get(i) ) {
	        	signChanges += 1;
	        }
	    }
	    return signChanges;
	}
	
	private static Comparator<List<Integer>> rowComparator =
		(one, two) -> Integer.compare(signChangeCount(one), signChangeCount(two));
	
	private static void display(List<List<Integer>> matrix) {
		for ( List<Integer> row : matrix ) {
			for ( int element : row ) {
				System.out.print(String.format("%3d", element));
			}
			System.out.println();
		}
		System.out.println();
	}	

}
Output:
Walsh matrix of order 2, Natural order:
  1  1  1  1
  1 -1  1 -1
  1  1 -1 -1
  1 -1 -1  1

Walsh matrix of order 4, Natural order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1

Walsh matrix of order 5, Natural order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1

Walsh matrix of order 2, Sequency order:
  1  1  1  1
  1  1 -1 -1
  1 -1 -1  1
  1 -1  1 -1

Walsh matrix of order 4, Sequency order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1

Walsh matrix of order 5, Sequency order:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1

Julia

kron is a builtin function in Julia. 

julia>using Plots

const w2 = [1 1; 1 -1]
walsh(k) = k < 2 ? w2 : kron(w2, walsh(k - 1))
countsignchanges(r) = count(i -> sign(r[i-1]) != sign(r[i[]]), 2:lastindex(r))
sequency(m) = sortslices(m, dims = 1, by = countsignchanges)

display(walsh(2))
display(walsh(3))
display(walsh(4))
display(sequency(walsh(3)))
display(sequency(walsh(4)))

subplots = [
    heatmap(
        (i ? sequency : identity)(walsh(n)),
        ylims = [0, 2^n + 1],
        xlims = [0, 2^n + 1],
        aspect_ratio = :equal,
        legend = false,
        axis = false,
        colormap = [:red, :forestgreen],
        yflip = true,
    ) for i = false:true, n = 3:5
]
plot(
    subplots...,
    plot_title = "Walsh, Natural Order" * "\u2007"^20 * "Walsh, Sequency Order",
    plot_titlefont = (9, "times"),
    layout = @layout [a b; c d; e f]
)
Output:
4×4 Matrix{Int64}:
 1   1   1   1
 1  -1   1  -1
 1   1  -1  -1
 1  -1  -1   1
8×8 Matrix{Int64}:
 1   1   1   1   1   1   1   1
 1  -1   1  -1   1  -1   1  -1
 1   1  -1  -1   1   1  -1  -1
 1  -1  -1   1   1  -1  -1   1
 1   1   1   1  -1  -1  -1  -1
 1  -1   1  -1  -1   1  -1   1
 1   1  -1  -1  -1  -1   1   1
 1  -1  -1   1  -1   1   1  -1
16×16 Matrix{Int64}:
 1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
 1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1
 1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1
 1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1
 1   1   1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1
 1  -1   1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1
 1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1   1   1
 1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1   1  -1
 1   1   1   1   1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1
 1  -1   1  -1   1  -1   1  -1  -1   1  -1   1  -1   1  -1   1
 1   1  -1  -1   1   1  -1  -1  -1  -1   1   1  -1  -1   1   1
 1  -1  -1   1   1  -1  -1   1  -1   1   1  -1  -1   1   1  -1
 1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1   1   1   1   1
 1  -1   1  -1  -1   1  -1   1  -1   1  -1   1   1  -1   1  -1
 1   1  -1  -1  -1  -1   1   1  -1  -1   1   1   1   1  -1  -1
 1  -1  -1   1  -1   1   1  -1  -1   1   1  -1   1  -1  -1   1
8×8 Matrix{Int64}:
 1   1   1   1   1   1   1   1
 1   1   1   1  -1  -1  -1  -1
 1   1  -1  -1  -1  -1   1   1
 1   1  -1  -1   1   1  -1  -1
 1  -1  -1   1   1  -1  -1   1
 1  -1  -1   1  -1   1   1  -1
 1  -1   1  -1  -1   1  -1   1
 1  -1   1  -1   1  -1   1  -1
16×16 Matrix{Int64}:
 1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
 1   1   1   1   1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1
 1   1   1   1  -1  -1  -1  -1  -1  -1  -1  -1   1   1   1   1
 1   1   1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1
 1   1  -1  -1  -1  -1   1   1   1   1  -1  -1  -1  -1   1   1
 1   1  -1  -1  -1  -1   1   1  -1  -1   1   1   1   1  -1  -1
 1   1  -1  -1   1   1  -1  -1  -1  -1   1   1  -1  -1   1   1
 1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1
 1  -1  -1   1   1  -1  -1   1   1  -1  -1   1   1  -1  -1   1
 1  -1  -1   1   1  -1  -1   1  -1   1   1  -1  -1   1   1  -1
 1  -1  -1   1  -1   1   1  -1  -1   1   1  -1   1  -1  -1   1
 1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1   1  -1
 1  -1   1  -1  -1   1  -1   1   1  -1   1  -1  -1   1  -1   1
 1  -1   1  -1  -1   1  -1   1  -1   1  -1   1   1  -1   1  -1
 1  -1   1  -1   1  -1   1  -1  -1   1  -1   1  -1   1  -1   1
 1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1   1  -1

Maxima

Using altern_kronecker as defined in Kronecker product task 

/* Function that attempts to implement recursion but only works for n when already called for every antecessor */
auxwalsh(n):=if n=1 then w[1]:matrix([1,1],[1,-1]) else 
block(w[2]:matrix([1,1,1,1],[1,-1,1,-1],[1,1,-1,-1],[1,-1,-1,1]),w[n]:altern_kronecker(w[1],w[n-1]),w[n])$

/* Function that guarantees an output for integer n */
walsh(n):=block(makelist(auxwalsh(i),i,1,n),last(%%))$

/* Examples */
walsh(4)$
wxdraw2d(palette = [red,gray,green], image(%,0,0,30,30))$

walsh(6)$
wxdraw2d(palette = [red,gray,green], image(%,0,0,30,30))$

Perl

#!/usr/bin/perl

use strict; # https://www.rosettacode.org/wiki/Walsh_matrix
use warnings;
use List::AllUtils qw( bundle_by pairwise nsort_by );

sub Kronecker
  {
  my ($ac, $bc) = map scalar($_->[0]->@*), my ($A, $B) = @_;
  return [ bundle_by { [ @_ ] } $ac * $bc, pairwise { $a * $b }
    @{[ map { map { ($_) x $bc } (@$_) x @$B } @$A ]}, # left side
    @{[ ( map { (@$_) x $ac } @$B ) x @$A ]} ];        # right side
  }

sub Walsh # Task - write a routine that, given k, returns Walsh of 2**k
  {
  my $k = shift;
  $k > 0 ? Kronecker [ [1,1],[1,-1] ], Walsh( $k - 1 ) : [[1]];
  }

for my $k ( 1, 3, 2, 4 ) # test code out of order just for fun
  {
  printf '%3d'x@$_ . "\n", @$_ for [], (my $w = Walsh($k))->@*, [];
  print nsort_by { scalar(() = /(.)\1*/g) }
    map { join '', (0, '_', '#')[@$_], "\n" } $w->@*;
  }
Output:

  1  1
  1 -1

__
_#

  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1

________
____####
__####__
__##__##
_##__##_
_##_#__#
_#_##_#_
_#_#_#_#

  1  1  1  1
  1 -1  1 -1
  1  1 -1 -1
  1 -1 -1  1

____
__##
_##_
_#_#

  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1

________________
________########
____########____
____####____####
__####____####__
__####__##____##
__##__####__##__
__##__##__##__##
_##__##__##__##_
_##__##_#__##__#
_##_#__##__#_##_
_##_#__#_##_#__#
_#_##_#__#_##_#_
_#_##_#_#_#__#_#
_#_#_#_##_#_#_#_
_#_#_#_#_#_#_#_#

Phix

Library: Phix/xpGUI
Library: Phix/online

You can run this online here. Use the keys '1'..'7' to change the order, limited to min 4 pixels per square, but you can resize/maximise the window, and the 's' key to toggle between natural and sequency order. 

with javascript_semantics
function walsh_matrix(integer n)
    sequence walsh = repeat(repeat(0,n),n)
    walsh[1, 1] = 1
    integer k = 1
    while k < n do
        for i=1 to k do
            for j=1 to k do
                integer wij = walsh[i, j]
                walsh[i+k, j  ] =  wij
                walsh[i  , j+k] =  wij
                walsh[i+k, j+k] = -wij
            end for
        end for
        k *= 2
    end while
    return walsh
end function

function sign_changes(sequence row)
    integer n = length(row)
    return sum(sq_eq(row[1..n-1],sq_mul(row[2..n],-1)))
end function

--/* -- console version:
for natural in {true,false} do
    for order in {2, 4, 5} do
        integer n = power(2,order)
        printf(1,"Walsh matrix - order %d (%d x %d), %s order:\n", {order, n, n, iff(natural?"natural":"sequency")})
        sequence w = walsh_matrix(n)
        if not natural then
            w = extract(w,custom_sort(apply(w,sign_changes),tagset(n)))
        end if
        pp(w,{pp_Nest,1,pp_IntFmt,"%2d",pp_Maxlen,132})
    end for
end for
--*/

include xpGUI.e

integer order = 2, natural = true

procedure redraw(gdx canvas)
    integer {w,h} = gGetAttribute(canvas,"SIZE"),
            mwh = min(w,h), n
    gCanvasRect(canvas,0,w,0,h,true,colour:=XPG_PARCHMENT,fillcolour:=XPG_PARCHMENT)
    while true do
        n = power(2,order)
        if n<=(floor(mwh/4)) then exit end if
        order -= 1
    end while
    string o = iff(natural?"natural":"sequency")
    gSetAttribute(gGetDialog(canvas),"TITLE","Walsh matrix order %d, %s order",{order,o})
    sequence m = walsh_matrix(n)
    if not natural then
        m = extract(m,custom_sort(apply(m,sign_changes),tagset(n)))
    end if
    integer s = floor(mwh/n), xm = floor((w-s*n)/2), ym = floor((h-s*n)/2)
    for i=1 to n do
        for j=1 to n do
            integer mij = m[i,j],
                    c = iff(mij=1?XPG_LIGHT_GREEN:XPG_RED),
                    x = (i-1)*s+xm,
                    y = (j-1)*s+ym
            gCanvasRect(canvas,x,x+s,y,y+s,true,colour:=XPG_BLACK,fillcolour:=c)
        end for
    end for
end procedure

function key_handler(gdx dlg, integer c)
    if c>='1' and c<='7' then
        order = c-'0' -- (may be limited within redraw())
        gRedraw(dlg)
	  return XPG_IGNORE
    elsif lower(c)='s' then
        natural = not natural
        gRedraw(dlg)
    end if
    return XPG_CONTINUE
end function

gdx canvas = gCanvas(redraw),
    dialog = gDialog(canvas,`gCanvas`,`SIZE=370x400`)
gCanvasSetBackground(canvas, XPG_PARCHMENT)
gSetHandler(dialog, `KEY`, key_handler)
gShow(dialog)
gMainLoop()

Raku

sub walsh (\m) { (map {$_?? -1 !! ' 1'}, map { :3(.base: 2) % 2 }, [X+&] ^2**m xx 2 ).batch: 2**m }

sub natural (@row) { Same }

sub sign-changes (@row) { sum (1..^@row).map: { 1 if @row[$_] !== @row[$_ - 1] } }

use SVG;

for &natural, 'natural', &sign-changes, 'sequency' -> &sort, $sort {
    for 2,4,5 -> $order {
        # ASCII text
        .put for "\nWalsh matrix - order $order ({exp($order,2)} x {exp($order,2)}), $sort order:", |walsh($order).sort: &sort;

        # SVG image
        my $side = 600;
        my $scale = $side / 2**$order;
        my $row = 0;
        my @blocks;
        my %C = ' 1' => '#0F0', '-1' => '#F00';

        for walsh($order).sort: &sort -> @row {
            my \x = $row++ * $scale;
            for @row.kv {
                my \y = $^k * $scale;
                @blocks.push: (:rect[:x(x),:y(y),:width($scale),:height($scale),:fill(%C{$^v})]);
            }
        }

        "walsh-matrix--order-{$order}--{$sort}-sort-order--raku.svg".IO.spurt:
          SVG.serialize(:svg[:width($side),:height($side),:stroke<black>,:stroke-width<1>,|@blocks])
    }
}
Output:
Walsh matrix - order 2 (4 x 4), natural order:
 1  1  1  1
 1 -1  1 -1
 1  1 -1 -1
 1 -1 -1  1

Walsh matrix - order 4 (16 x 16), natural order:
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1

Walsh matrix - order 5 (32 x 32), natural order:
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1

Walsh matrix - order 2 (4 x 4), sequency order:
 1  1  1  1
 1  1 -1 -1
 1 -1 -1  1
 1 -1  1 -1

Walsh matrix - order 4 (16 x 16), sequency order:
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1

Walsh matrix - order 5 (32 x 32), sequency order:
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1
 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1
 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1
 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1
 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1
 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1
 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1
 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1
 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1
 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1
 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1
 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1
Natural order Sequency order

Wren

Library: Wren-matrix
Library: Wren-fmt

Wren-cli

Text mode version. 

import "./matrix" for Matrix
import "./fmt" for Fmt

var walshMatrix = Fn.new { |n|
    var walsh = Matrix.new(n, n, 0)
    walsh[0, 0] = 1
    var k = 1
    while (k < n) {
        for (i in 0...k) {
            for (j in 0...k) {
                walsh[i+k, j]   =  walsh[i, j]
                walsh[i, j+k]   =  walsh[i, j]
                walsh[i+k, j+k] = -walsh[i, j]
            }
        }
        k = k + k
    }
    return walsh
}

var signChanges = Fn.new { |row|
    var n = row.count
    var sc = 0
    for (i in 1...n) {
        if (row[i-1] == -row[i]) sc = sc + 1
    }
    return sc
}

var walshCache = {} // to avoid calculating the Walsh matrix twice

for (order in [2, 4, 5]) {
    var n = 1 << order
    Fmt.print("Walsh matrix - order $d ($d x $d), natural order:", order, n, n)
    var w = walshMatrix.call(n)
    walshCache[order] = w
    Fmt.mprint(w, 2, 0, "|", true)
    System.print()
}

for (order in [2, 4, 5]) {
    var n = 1 << order
    Fmt.print("Walsh matrix - order $d ($d x $d), sequency order:", order, n, n)
    var rows = walshCache[order].toList
    rows.sort { |r1, r2| signChanges.call(r1) < signChanges.call(r2) }
    Fmt.mprint(rows, 2, 0, "|", true)
    System.print()
}
Output:
Walsh matrix - order 2 (4 x 4), natural order:
| 1  1  1  1|
| 1 -1  1 -1|
| 1  1 -1 -1|
| 1 -1 -1  1|

Walsh matrix - order 4 (16 x 16), natural order:
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1|

Walsh matrix - order 5 (32 x 32), natural order:
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1|
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1|

Walsh matrix - order 2 (4 x 4), sequency order:
| 1  1  1  1|
| 1  1 -1 -1|
| 1 -1 -1  1|
| 1 -1  1 -1|

Walsh matrix - order 4 (16 x 16), sequency order:
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1|

Walsh matrix - order 5 (32 x 32), sequency order:
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
| 1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1|
| 1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1|
| 1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1  1  1  1  1 -1 -1 -1 -1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1|
| 1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1|
| 1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1|
| 1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1|
| 1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1|
| 1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1|
| 1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1|
| 1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1|
| 1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1|
| 1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1|
| 1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1|
| 1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1|
| 1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1  1 -1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1|
| 1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1|

DOME

Library: DOME
Library: Wren-polygon

Image mode version. 

import "dome" for Window
import "input" for Keyboard
import "graphics" for Canvas, Color
import "./matrix" for Matrix
import "./polygon" for Square

var walshMatrix = Fn.new { |n|
    var walsh = Matrix.new(n, n, 0)
    walsh[0, 0] = 1
    var k = 1
    while (k < n) {
        for (i in 0...k) {
            for (j in 0...k) {
                walsh[i+k, j]   =  walsh[i, j]
                walsh[i, j+k]   =  walsh[i, j]
                walsh[i+k, j+k] = -walsh[i, j]
            }
        }
        k = k + k
    }
    return walsh
}

var signChanges = Fn.new { |row|
    var n = row.count
    var sc = 0
    for (i in 1...n) {
        if (row[i-1] == -row[i]) sc = sc + 1
    }
    return sc
}

var WalshNaturalCache = {}
var WalshSequencyCache = {}

for (order in [2, 4, 5]) {
    var n = 1 << order
    var w = walshMatrix.call(n).toList
    WalshNaturalCache[order] = w
}

for (order in [2, 4, 5]) {
    var rows = WalshNaturalCache[order].toList
    rows.sort { |r1, r2| signChanges.call(r1) < signChanges.call(r2) }
    WalshSequencyCache[order] = rows
}

class WalshMatrix {
    construct new() {
        Window.title = "Walsh Matrix"
        Window.resize(1020, 750)
        Canvas.resize(1020, 750)
        var bc = Color.black
        for (natural in [true, false]) {
            if (natural) {
                Canvas.print("NATURAL ORDERING", 450, 10, Color.blue)
            } else {
                Canvas.print("SEQUENCY ORDERING", 450, 400, Color.blue)
            }
            var z = 10
            for (order in [2, 4, 5]) {
                var y = natural ? 30 : 420
                var mat = natural ? WalshNaturalCache[order] : WalshSequencyCache[order]
                var n = 1 << order
                var size = 320 / n
                for (row in mat) {
                    var x = z
                    for (i in row) {
                        var fc = (i == 1) ? Color.green : Color.red
                        var sq = Square.new(x, y, size)
                        sq.drawfill(fc, bc)
                        x = x + size
                    }
                    y = y + size
                }
                z = z + 340
            }
        }
    }

    init() {}

    update() {}

    draw(alpha) {}
}

var Game = WalshMatrix.new()
Retrieved from "https://rosettacode.org/wiki/Walsh_matrix?oldid=358311"