Matrix-exponentiation operator

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Task
Matrix-exponentiation operator
You are encouraged to solve this task according to the task description, using any language you may know.
Most programming languages have a built-in implementation of exponentiation for integers and reals only.

Demonstrate how to implement matrix exponentiation as an operator.

Contents

[edit] Ada

This is a generic solution for any natural power exponent. It will work with any type that has +,*, additive and multiplicative 0s. The implementation factors out powers A2n:

with Ada.Text_IO;  use Ada.Text_IO;
 
procedure Test_Matrix is
generic
type Element is private;
Zero : Element;
One  : Element;
with function "+" (A, B : Element) return Element is <>;
with function "*" (A, B : Element) return Element is <>;
with function Image (X : Element) return String is <>;
package Matrices is
type Matrix is array (Integer range <>, Integer range <>) of Element;
function "*" (A, B : Matrix) return Matrix;
function "**" (A : Matrix; Power : Natural) return Matrix;
procedure Put (A : Matrix);
end Matrices;
 
package body Matrices is
function "*" (A, B : Matrix) return Matrix is
R  : Matrix (A'Range (1), B'Range (2));
Sum : Element := Zero;
begin
for I in R'Range (1) loop
for J in R'Range (2) loop
Sum := Zero;
for K in A'Range (2) loop
Sum := Sum + A (I, K) * B (K, J);
end loop;
R (I, J) := Sum;
end loop;
end loop;
return R;
end "*";
 
function "**" (A : Matrix; Power : Natural) return Matrix is
begin
if Power = 1 then
return A;
end if;
declare
R : Matrix (A'Range (1), A'Range (2)) := (others => (others => Zero));
P : Matrix  := A;
E : Natural := Power;
begin
for I in P'Range (1) loop -- R is identity matrix
R (I, I) := One;
end loop;
if E = 0 then
return R;
end if;
loop
if E mod 2 /= 0 then
R := R * P;
end if;
E := E / 2;
exit when E = 0;
P := P * P;
end loop;
return R;
end;
end "**";
 
procedure Put (A : Matrix) is
begin
for I in A'Range (1) loop
for J in A'Range (1) loop
Put (Image (A (I, J)));
end loop;
New_Line;
end loop;
end Put;
end Matrices;
 
package Integer_Matrices is new Matrices (Integer, 0, 1, Image => Integer'Image);
use Integer_Matrices;
 
M : Matrix (1..2, 1..2) := ((3,2),(2,1));
begin
Put_Line ("M ="); Put (M);
Put_Line ("M**0 ="); Put (M**0);
Put_Line ("M**1 ="); Put (M**1);
Put_Line ("M**2 ="); Put (M**2);
Put_Line ("M*M ="); Put (M*M);
Put_Line ("M**3 ="); Put (M**3);
Put_Line ("M*M*M ="); Put (M*M*M);
Put_Line ("M**4 ="); Put (M**4);
Put_Line ("M*M*M*M ="); Put (M*M*M*M);
Put_Line ("M**10 ="); Put (M**10);
Put_Line ("M*M*M*M*M*M*M*M*M*M ="); Put (M*M*M*M*M*M*M*M*M*M);
end Test_Matrix;

Sample output:

M =
 3 2
 2 1
M**0 =
 1 0
 0 1
M**1 =
 3 2
 2 1
M**2 =
 13 8
 8 5
M*M =
 13 8
 8 5
M**3 =
 55 34
 34 21
M*M*M =
 55 34
 34 21
M**4 =
 233 144
 144 89
M*M*M*M =
 233 144
 144 89
M**10 =
 1346269 832040
 832040 514229
M*M*M*M*M*M*M*M*M*M =
 1346269 832040
 832040 514229

The following program implements exponentiation of a square Hermitian complex matrix by any complex power. The limitation to be Hermitian is not essential and comes for the limitation of the standard Ada linear algebra library.

with Ada.Text_IO;                  use Ada.Text_IO;
with Ada.Complex_Text_IO; use Ada.Complex_Text_IO;
with Ada.Numerics.Complex_Types; use Ada.Numerics.Complex_Types;
with Ada.Numerics.Real_Arrays; use Ada.Numerics.Real_Arrays;
with Ada.Numerics.Complex_Arrays; use Ada.Numerics.Complex_Arrays;
with Ada.Numerics.Complex_Elementary_Functions; use Ada.Numerics.Complex_Elementary_Functions;
 
procedure Test_Matrix is
function "**" (A : Complex_Matrix; Power : Complex) return Complex_Matrix is
L  : Real_Vector (A'Range (1));
X  : Complex_Matrix (A'Range (1), A'Range (2));
R  : Complex_Matrix (A'Range (1), A'Range (2));
RL : Complex_Vector (A'Range (1));
begin
Eigensystem (A, L, X);
for I in L'Range loop
RL (I) := (L (I), 0.0) ** Power;
end loop;
for I in R'Range (1) loop
for J in R'Range (2) loop
declare
Sum : Complex := (0.0, 0.0);
begin
for K in RL'Range (1) loop
Sum := Sum + X (K, I) * RL (K) * X (K, J);
end loop;
R (I, J) := Sum;
end;
end loop;
end loop;
return R;
end "**";
procedure Put (A : Complex_Matrix) is
begin
for I in A'Range (1) loop
for J in A'Range (1) loop
Put (A (I, J));
end loop;
New_Line;
end loop;
end Put;
M : Complex_Matrix (1..2, 1..2) := (((3.0,0.0),(2.0,1.0)),((2.0,-1.0),(1.0,0.0)));
begin
Put_Line ("M ="); Put (M);
Put_Line ("M**0 ="); Put (M**(0.0,0.0));
Put_Line ("M**1 ="); Put (M**(1.0,0.0));
Put_Line ("M**0.5 ="); Put (M**(0.5,0.0));
end Test_Matrix;

This solution is not tested, because the available version of GNAT GPL Ada compiler (20070405-41) does not provide an implementation of the standard library.

[edit] ALGOL 68

Works with: ALGOL 68 version Revision 1 - no extensions to language used.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.

File: Matrix_algebra.a68

INT default upb=3;
MODE VEC = [default upb]COSCAL;
MODE MAT = [default upb,default upb]COSCAL;
 
OP * = (VEC a,b)COSCAL: (
COSCAL result:=0;
FOR i FROM LWB a TO UPB a DO result+:= a[i]*b[i] OD;
result
);
 
OP * = (VEC a, MAT b)VEC: ( # overload vec times matrix #
[2 LWB b:2 UPB b]COSCAL result;
FOR j FROM 2 LWB b TO 2 UPB b DO result[j]:=a*b[,j] OD;
result
);
 
OP * = (MAT a, b)MAT: ( # overload matrix times matrix #
[LWB a:UPB a, 2 LWB b:2 UPB b]COSCAL result;
FOR k FROM LWB result TO UPB result DO result[k,]:=a[k,]*b OD;
result
);
 
OP IDENTITY = (INT upb)MAT:(
[upb,upb] COSCAL out;
FOR i TO upb DO
FOR j TO upb DO
out[i,j]:= ( i=j |1|0)
OD
OD;
out
);
File: Matrix-exponentiation_operator.a68
OP ** = (MAT base, INT exponent)MAT: (
BITS binary exponent:=BIN exponent ;
MAT out := IF bits width ELEM binary exponent THEN base ELSE IDENTITY UPB base FI;
MAT sq:=base;
 
WHILE
binary exponent := binary exponent SHR 1;
binary exponent /= BIN 0
DO
sq := sq * sq;
IF bits width ELEM binary exponent THEN out := out * sq FI
OD;
out
);
File: test_Matrix-exponentiation_operator.a68
#!/usr/local/bin/a68g --script #
 
MODE COSCAL = COMPL;
PR READ "Matrix_algebra.a68" PR
PR READ "Matrix-exponentiation_operator.a68" PR
 
PROC compl mat printf= (FORMAT scal fmt, MAT m)VOID:(
FORMAT
vec math = $n(2 UPB m)(f(scal fmt)"&")$,
mat math = $"<math>\begin{bmat}"ln(UPB m)(xxf(vec fmt)"\\"l)"\end{bmat}</math>"$,
vec fmt = $"("n(2 UPB m-1)(f(scal fmt)",")f(scal fmt)")"$,
mat fmt = $x"("n(UPB m-1)(f(vec fmt)","lxx)f(vec fmt)");"$;
# finally print the result #
printf((mat fmt,m))
);
 
FORMAT scal fmt = $-d.dddd,+d.dddd"i"$; # width of 4, with no leading '+' sign, 1 decimals #
MAT mat=((sqrt(0.5)I0 , sqrt(0.5)I0 , 0I0),
( 0I-sqrt(0.5), 0Isqrt(0.5), 0I0),
( 0I0 , 0I0 , 0I1))
 
printf(($" mat ** "g(0)":"l$,24));
compl mat printf(scal fmt, mat**24);
print(newline)

Output:

 mat ** 24:
 (( 1.0000+0.0000i, 0.0000+0.0000i, 0.0000+0.0000i),
  ( 0.0000+0.0000i, 1.0000+0.0000i, 0.0000+0.0000i),
  ( 0.0000+0.0000i, 0.0000+0.0000i, 1.0000+0.0000i));

[edit] BBC BASIC

      DIM matrix(1,1), output(1,1)
matrix() = 3, 2, 2, 1
 
FOR power% = 0 TO 9
PROCmatrixpower(matrix(), output(), power%)
PRINT "matrix()^" ; power% " = "
FOR row% = 0 TO DIM(output(), 1)
FOR col% = 0 TO DIM(output(), 2)
PRINT output(row%,col%);
NEXT
PRINT
NEXT row%
NEXT power%
END
 
DEF PROCmatrixpower(src(), dst(), pow%)
LOCAL i%
dst() = 0
FOR i% = 0 TO DIM(dst(), 1) : dst(i%,i%) = 1 : NEXT
IF pow% THEN
FOR i% = 1 TO pow%
dst() = dst() . src()
NEXT
ENDIF
ENDPROC

Output:

matrix()^0 =
         1         0
         0         1
matrix()^1 =
         3         2
         2         1
matrix()^2 =
        13         8
         8         5
matrix()^3 =
        55        34
        34        21
matrix()^4 =
       233       144
       144        89
matrix()^5 =
       987       610
       610       377
matrix()^6 =
      4181      2584
      2584      1597
matrix()^7 =
     17711     10946
     10946      6765
matrix()^8 =
     75025     46368
     46368     28657
matrix()^9 =
    317811    196418
    196418    121393

[edit] C

C doesn't support classes or allow operator overloading. The following is code that defines a function, SquareMtxPower that will raise a matrix to a positive integer power.

#include <math.h>
#include <stdio.h>
#include <stdlib.h>
 
typedef struct squareMtxStruct {
int dim;
double *cells;
double **m;
} *SquareMtx;
 
/* function for initializing row r of a new matrix */
typedef void (*FillFunc)( double *cells, int r, int dim, void *ff_data);
 
SquareMtx NewSquareMtx( int dim, FillFunc fillFunc, void *ff_data )
{
SquareMtx sm = malloc(sizeof(struct squareMtxStruct));
if (sm) {
int rw;
sm->dim = dim;
sm->cells = malloc(dim*dim * sizeof(double));
sm->m = malloc( dim * sizeof(double *));
if ((sm->cells != NULL) && (sm->m != NULL)) {
for (rw=0; rw<dim; rw++) {
sm->m[rw] = sm->cells + dim*rw;
fillFunc( sm->m[rw], rw, dim, ff_data );
}
}
else {
free(sm->m);
free(sm->cells);
free(sm);
printf("Square Matrix allocation failure\n");
return NULL;
}
}
else {
printf("Malloc failed for square matrix\n");
}
return sm;
}
 
void ffMatxSquare( double *cells, int rw, int dim, SquareMtx m0 )
{
int col, ix;
double sum;
double *m0rw = m0->m[rw];
 
for (col = 0; col < dim; col++) {
sum = 0.0;
for (ix=0; ix<dim; ix++)
sum += m0rw[ix] * m0->m[ix][col];
cells[col] = sum;
}
}
 
void ffMatxMulply( double *cells, int rw, int dim, SquareMtx mplcnds[] )
{
SquareMtx mleft = mplcnds[0];
SquareMtx mrigt = mplcnds[1];
double sum;
double *m0rw = mleft->m[rw];
int col, ix;
 
for (col = 0; col < dim; col++) {
sum = 0.0;
for (ix=0; ix<dim; ix++)
sum += m0rw[ix] * mrigt->m[ix][col];
cells[col] = sum;
}
}
 
void MatxMul( SquareMtx mr, SquareMtx left, SquareMtx rigt)
{
int rw;
SquareMtx mplcnds[2];
mplcnds[0] = left; mplcnds[1] = rigt;
 
for (rw = 0; rw < left->dim; rw++)
ffMatxMulply( mr->m[rw], rw, left->dim, mplcnds);
}
 
void ffIdentity( double *cells, int rw, int dim, void *v )
{
int col;
for (col=0; col<dim; col++) cells[col] = 0.0;
cells[rw] = 1.0;
}
void ffCopy(double *cells, int rw, int dim, SquareMtx m1)
{
int col;
for (col=0; col<dim; col++) cells[col] = m1->m[rw][col];
}
 
void FreeSquareMtx( SquareMtx m )
{
free(m->m);
free(m->cells);
free(m);
}
 
SquareMtx SquareMtxPow( SquareMtx m0, int exp )
{
SquareMtx v0 = NewSquareMtx(m0->dim, ffIdentity, NULL);
SquareMtx v1 = NULL;
SquareMtx base0 = NewSquareMtx( m0->dim, ffCopy, m0);
SquareMtx base1 = NULL;
SquareMtx mplcnds[2], t;
 
while (exp) {
if (exp % 2) {
if (v1)
MatxMul( v1, v0, base0);
else {
mplcnds[0] = v0; mplcnds[1] = base0;
v1 = NewSquareMtx(m0->dim, ffMatxMulply, mplcnds);
}
{t = v0; v0=v1; v1 = t;}
}
if (base1)
MatxMul( base1, base0, base0);
else
base1 = NewSquareMtx( m0->dim, ffMatxSquare, base0);
t = base0; base0 = base1; base1 = t;
exp = exp/2;
}
if (base0) FreeSquareMtx(base0);
if (base1) FreeSquareMtx(base1);
if (v1) FreeSquareMtx(v1);
return v0;
}
 
FILE *fout;
void SquareMtxPrint( SquareMtx mtx, const char *mn )
{
int rw, col;
int d = mtx->dim;
 
fprintf(fout, "%s dim:%d =\n", mn, mtx->dim);
 
for (rw=0; rw<d; rw++) {
fprintf(fout, " |");
for(col=0; col<d; col++)
fprintf(fout, "%8.5f ",mtx->m[rw][col] );
fprintf(fout, " |\n");
}
fprintf(fout, "\n");
}
 
void fillInit( double *cells, int rw, int dim, void *data)
{
double theta = 3.1415926536/6.0;
double c1 = cos( theta);
double s1 = sin( theta);
 
switch(rw) {
case 0:
cells[0]=c1; cells[1]=s1; cells[2]=0.0;
break;
case 1:
cells[0]=-s1; cells[1]=c1; cells[2]=0;
break;
case 2:
cells[0]=0.0; cells[1]=0.0; cells[2]=1.0;
break;
}
}
 
int main()
{
SquareMtx m0 = NewSquareMtx( 3, fillInit, NULL);
SquareMtx m1 = SquareMtxPow( m0, 5);
SquareMtx m2 = SquareMtxPow( m0, 9);
SquareMtx m3 = SquareMtxPow( m0, 2);
 
// fout = stdout;
fout = fopen("matrx_exp.txt", "w");
SquareMtxPrint(m0, "m0"); FreeSquareMtx(m0);
SquareMtxPrint(m1, "m0^5"); FreeSquareMtx(m1);
SquareMtxPrint(m2, "m0^9"); FreeSquareMtx(m2);
SquareMtxPrint(m3, "m0^2"); FreeSquareMtx(m3);
fclose(fout);
 
return 0;
}

Output:

m0 dim:3 =
 | 0.86603  0.50000  0.00000  |
 |-0.50000  0.86603  0.00000  |
 | 0.00000  0.00000  1.00000  |

m0^5 dim:3 =
 |-0.86603  0.50000  0.00000  |
 |-0.50000 -0.86603  0.00000  |
 | 0.00000  0.00000  1.00000  |

m0^9 dim:3 =
 | 0.00000 -1.00000  0.00000  |
 | 1.00000  0.00000  0.00000  |
 | 0.00000  0.00000  1.00000  |

m0^2 dim:3 =
 | 0.50000  0.86603  0.00000  |
 |-0.86603  0.50000  0.00000  |
 | 0.00000  0.00000  1.00000  |

[edit] C++

This is an implementation in C++.

#include <complex>
#include <cmath>
#include <iostream>
using namespace std;
 
template<int MSize = 3, class T = complex<double> >
class SqMx {
typedef T Ax[MSize][MSize];
typedef SqMx<MSize, T> Mx;
 
private:
Ax a;
SqMx() { }
 
public:
SqMx(const Ax &_a) { // constructor with pre-defined array
for (int r = 0; r < MSize; r++)
for (int c = 0; c < MSize; c++)
a[r][c] = _a[r][c];
}
 
static Mx identity() {
Mx m;
for (int r = 0; r < MSize; r++)
for (int c = 0; c < MSize; c++)
m.a[r][c] = (r == c ? 1 : 0);
return m;
}
 
friend ostream &operator<<(ostream& os, const Mx &p)
{ // ugly print
for (int i = 0; i < MSize; i++) {
for (int j = 0; j < MSize; j++)
os << p.a[i][j] << ",";
os << endl;
}
return os;
}
 
Mx operator*(const Mx &b) {
Mx d;
for (int r = 0; r < MSize; r++)
for (int c = 0; c < MSize; c++) {
d.a[r][c] = 0;
for (int k = 0; k < MSize; k++)
d.a[r][c] += a[r][k] * b.a[k][c];
}
return d;
}

This is the task part.

  // C++ does not have a ** operator, instead, ^ (bitwise Xor) is used.
Mx operator^(int n) {
if (n < 0)
throw "Negative exponent not implemented";
 
Mx d = identity();
for (Mx sq = *this; n > 0; sq = sq * sq, n /= 2)
if (n % 2 != 0)
d = d * sq;
return d;
}
};
 
typedef SqMx<> M3;
typedef complex<double> creal;
 
int main() {
double q = sqrt(0.5);
creal array[3][3] =
{{creal(q, 0), creal(q, 0), creal(0, 0)},
{creal(0, -q), creal(0, q), creal(0, 0)},
{creal(0, 0), creal(0, 0), creal(0, 1)}};
M3 m(array);
 
cout << "m ^ 23=" << endl
<< (m ^ 23) << endl;
 
return 0;
}

Output:

m ^ 23=
(0.707107,0),(0,0.707107),(0,0),
(0.707107,0),(0,-0.707107),(0,0),
(0,0),(0,0),(0,-1),

An alternative way would be to implement operator*= and conversion from number (giving multiples of the identity matrix) for the matrix and use the generic code from Exponentiation operator#C++ with support for negative exponents removed (or alternatively, implement matrix inversion as well, implement /= in terms of it, and use the generic code unchanged). Note that the algorithm used there is much faster as well.

[edit] Common Lisp

This Common Lisp implementation uses 2D Arrays to represent matrices, and checks to make sure that the arrays are the right dimensions for multiplication and square for exponentiation.

(defun multiply-matrices (matrix-0 matrix-1)
"Takes two 2D arrays and returns their product, or an error if they cannot be multiplied"
(let* ((m0-dims (array-dimensions matrix-0))
(m1-dims (array-dimensions matrix-1))
(m0-dim (length m0-dims))
(m1-dim (length m1-dims)))
(if (or (/= 2 m0-dim) (/= 2 m1-dim))
(error "Array given not a matrix")
(let ((m0-rows (car m0-dims))
(m0-cols (cadr m0-dims))
(m1-rows (car m1-dims))
(m1-cols (cadr m1-dims)))
(if (/= m0-cols m1-rows)
(error "Incompatible dimensions")
(do ((rarr (make-array (list m0-rows m1-cols)
:initial-element 0) rarr)
(n 0 (if (= n (1- m0-cols)) 0 (1+ n)))
(cc 0 (if (= n (1- m0-cols))
(if (/= cc (1- m1-cols))
(1+ cc) 0) cc))
(cr 0 (if (and (= (1- m0-cols) n)
(= (1- m1-cols) cc))
(1+ cr)
cr)))
((= cr m0-rows) rarr)
(setf (aref rarr cr cc)
(+ (aref rarr cr cc)
(* (aref matrix-0 cr n)
(aref matrix-1 n cc))))))))))
 
(defun matrix-identity (dim)
"Creates a new identity matrix of size dim*dim"
(do ((rarr (make-array (list dim dim)
:initial-element 0) rarr)
(n 0 (1+ n)))
((= n dim) rarr)
(setf (aref rarr n n) 1)))
 
(defun matrix-expt (matrix exp)
"Takes the first argument (a matrix) and multiplies it by itself exp times"
(let* ((m-dims (array-dimensions matrix))
(m-rows (car m-dims))
(m-cols (cadr m-dims)))
(cond
((/= m-rows m-cols) (error "Non-square matrix"))
((zerop exp) (matrix-identity m-rows))
((= 1 exp) (do ((rarr (make-array (list m-rows m-cols)) rarr)
(cc 0 (if (= cc (1- m-cols))
0
(1+ cc)))
(cr 0 (if (= cc (1- m-cols))
(1+ cr)
cr)))
((= cr m-rows) rarr)
(setf (aref rarr cr cc) (aref matrix cr cc))))
((zerop (mod exp 2)) (let ((me2 (matrix-expt matrix (/ exp 2))))
(multiply-matrices me2 me2)))
(t (let ((me2 (matrix-expt matrix (/ (1- exp) 2))))
(multiply-matrices matrix (multiply-matrices me2 me2)))))))

Output (note that this lisp implementation uses single-precision floats for decimals by default). We can also use rationals:

CL-USER> (setf 5x5-matrix
               (make-array '(5 5)
                           :initial-contents
                           '((0    1 -1   -2    2)
                             (0.4  4  3.2 -3  -10)
                             (4.5 -2  0.5  1    7)
                             (10   1  0    1.5 -2)
                             (4    5 -3   -2    1))))
#2A((0 1 -1 -2 2)
    (0.4 4 3.2 -3 -10)
    (4.5 -2 0.5 1 7)
    (10 1 0 1.5 -2)
    (4 5 -3 -2 1))
CL-USER> (matrix-expt 5x5-matrix 3)
#2A((-163.25 -19.5 92.25 -7.5999985 -184.3)
    (156.6 -412.09998 0.7999954 331.45 597.4)
    (-129.82501 401.25 -66.975 -302.55 -390.15)
    (-148.9 39.25 -5.200001 -67.225006 -7.300003)
    (-495.05 -231.5 310.85 33.0 -328.5))
CL-USER> (setf 4x4-matrix
               (make-array '(4 4)
                           :initial-contents
                           '(( 1/2 -1/2  4    8)
                             (-3/4  7/3  8/5 -2)
                             (-5   17   20/3 -5/2)
                             ( 3/2 -1   -7/3  6))))                            
#2A((1/2 -1/2 4 8) (-3/4 7/3 8/5 -2) (-5 17 20/3 -5/2) (3/2 -1 -7/3 6))
CL-USER> (matrix-expt 4x4-matrix 3)
#2A((-233/8 182723/720 757/30 353/6)
    (-73517/480 838241/2160 77789/450 -67537/180)
    (-5315/9 66493/45 90883/135 -54445/36)
    (37033/144 -27374/45 -15515/54 12109/18))

[edit] Chapel

This uses the '*' operator for arrays as defined in Matrix_multiplication#Chapel

proc **(a, e) {
// create result matrix of same dimensions
var r:[a.domain] a.eltType;
// and initialize to identity matrix
forall ij in r.domain do
r(ij) = if ij(1) == ij(2) then 1 else 0;
 
for 1..e do
r *= a;
 
return r;
}

Usage example (like Perl):

var m:[1..3, 1..3] int;
m(1,1) = 1; m(1,2) = 2; m(1,3) = 0;
m(2,1) = 0; m(2,2) = 3; m(2,3) = 1;
m(3,1) = 1; m(3,2) = 0; m(3,3) = 0;
 
config param n = 10;
 
for i in 0..n do {
writeln("Order ", i);
writeln(m ** i, "\n");
}
Output:
Order 0
1 0 0
0 1 0
0 0 1

Order 1
1 2 0
0 3 1
1 0 0

Order 2
1 8 2
1 9 3
1 2 0

Order 3
3 26 8
4 29 9
1 8 2

Order 4
11 84 26
13 95 29
3 26 8

Order 5
37 274 84
42 311 95
11 84 26

Order 6
121 896 274
137 1017 311
37 274 84

Order 7
395 2930 896
448 3325 1017
121 896 274

Order 8
1291 9580 2930
1465 10871 3325
395 2930 896

Order 9
4221 31322 9580
4790 35543 10871
1291 9580 2930

Order 10
13801 102408 31322
15661 116209 35543
4221 31322 9580

[edit] D

import std.stdio, std.string, std.math, std.array;
 
struct SquareMat(T = creal) {
public static string fmt = "%8.3f";
private alias TM = T[][];
private TM a;
 
public this(in size_t side) pure nothrow
in {
assert(side > 0);
} body {
a = new TM(side, side);
}
 
public this(in TM m) pure nothrow
in {
assert(!m.empty);
foreach (const row; m)
assert(m.length == m[0].length);
} body {
a.length = m.length;
foreach (immutable i, const row; m)
//a[i] = row.dup; // Not nothrow.
a[i] = row ~ []; // Slower.
}
 
string toString() const {
return format("<%(%(" ~ fmt ~ ", %)\n %)>", a);
}
 
public static SquareMat identity(in size_t side) pure nothrow {
SquareMat m;
m.a.length = side;
foreach (immutable r, ref row; m.a) {
row.length = side;
foreach (immutable c; 0 .. side)
row[c] = cast(T)(r == c ? 1 : 0);
}
return m;
}
 
public SquareMat opBinary(string op:"*")(in SquareMat other)
const pure nothrow in {
assert (a.length == other.a.length);
} body {
immutable size_t side = other.a.length;
SquareMat d;
d.a = new TM(side, side);
foreach (immutable r; 0 .. side)
foreach (immutable c; 0 .. side) {
d.a[r][c] = cast(T)0;
foreach (immutable k, immutable ark; a[r])
d.a[r][c] += ark * other.a[k][c];
}
return d;
}
 
// This is the task part ---------------
public SquareMat opBinary(string op:"^^")(int n) const pure nothrow
in {
assert(n >= 0, "Negative exponent not implemented.");
} body {
auto sq = SquareMat(this.a);
auto d = SquareMat.identity(a.length);
for (; n > 0; sq = sq * sq, n >>= 1)
if (n & 1)
d = d * sq;
return d;
}
}
 
void main() {
alias M = SquareMat!();
immutable real q = sqrt(0.5);
immutable m = M([[ q + 0*1.0Li, q + 0*1.0Li, 0.0L + 0.0Li],
[0.0L - q*1.0Li, 0.0L + q*1.0Li, 0.0L + 0.0Li],
[0.0L + 0.0Li, 0.0L + 0.0Li, 0.0L + 1.0Li]]);
M.fmt = "%5.2f";
foreach (immutable p; [0, 1, 23, 24])
writefln("m ^^ %d =\n%s", p, m ^^ p);
}
Output:
m ^^ 0 =
< 1.00+ 0.00i,  0.00+ 0.00i,  0.00+ 0.00i
  0.00+ 0.00i,  1.00+ 0.00i,  0.00+ 0.00i
  0.00+ 0.00i,  0.00+ 0.00i,  1.00+ 0.00i>
m ^^ 1 =
< 0.71+ 0.00i,  0.71+ 0.00i,  0.00+ 0.00i
  0.00+-0.71i,  0.00+ 0.71i,  0.00+ 0.00i
  0.00+ 0.00i,  0.00+ 0.00i,  0.00+ 1.00i>
m ^^ 23 =
< 0.71+ 0.00i,  0.00+ 0.71i,  0.00+ 0.00i
  0.71+ 0.00i,  0.00+-0.71i,  0.00+ 0.00i
  0.00+ 0.00i,  0.00+ 0.00i,  0.00+-1.00i>
m ^^ 24 =
< 1.00+ 0.00i,  0.00+ 0.00i,  0.00+ 0.00i
  0.00+ 0.00i,  1.00+ 0.00i,  0.00+ 0.00i
  0.00+ 0.00i,  0.00+ 0.00i,  1.00+ 0.00i>

[edit] Factor

There is already a built-in word (m^n) that implements exponentiation. Here is a simple and less efficient implementation.

USING: kernel math math.matrices sequences ;
 
: my-m^n ( m n -- m' )
dup 0 < [ "no negative exponents" throw ] [
[ drop length identity-matrix ]
[ swap '[ _ m. ] times ] 2bi
] if ;
( scratchpad ) { { 3 2 } { 2 1 } } 0 my-m^n .
{ { 1 0 } { 0 1 } }
( scratchpad ) { { 3 2 } { 2 1 } } 4 my-m^n .
{ { 233 144 } { 144 89 } }

[edit] Fortran

Works with: Fortran version 90 and later
module matmod
implicit none
 
! Overloading the ** operator does not work because the compiler cannot
! differentiate between matrix exponentiation and the elementwise raising
! of an array to a power therefore we define a new operator
interface operator (.matpow.)
module procedure matrix_exp
end interface
 
contains
 
function matrix_exp(m, n) result (res)
real, intent(in) :: m(:,:)
integer, intent(in) :: n
real :: res(size(m,1),size(m,2))
integer :: i
 
if(n == 0) then
res = 0
do i = 1, size(m,1)
res(i,i) = 1
end do
return
end if
 
res = m
do i = 2, n
res = matmul(res, m)
end do
 
end function matrix_exp
end module matmod
 
program Matrix_exponentiation
use matmod
implicit none
 
integer, parameter :: n = 3
real, dimension(n,n) :: m1, m2
integer :: i, j
 
m1 = reshape((/ (i, i = 1, n*n) /), (/ n, n /), order = (/ 2, 1 /))
 
do i = 0, 4
m2 = m1 .matpow. i
do j = 1, size(m2,1)
write(*,*) m2(j,:)
end do
write(*,*)
end do
 
end program Matrix_exponentiation

Output

      1.00000         0.00000         0.00000
      0.00000         1.00000         0.00000
      0.00000         0.00000         1.00000
 
      1.00000         2.00000         3.00000
      4.00000         5.00000         6.00000
      7.00000         8.00000         9.00000
 
      30.0000         36.0000         42.0000
      66.0000         81.0000         96.0000
      102.000         126.000         150.000
 
      468.000         576.000         684.000
      1062.00         1305.00         1548.00
      1656.00         2034.00         2412.00
 
      7560.00         9288.00         11016.0
      17118.0         21033.0         24948.0
      26676.0         32778.0         38880.0

[edit] GAP

# Matrix exponentiation is built-in
A := [[0 , 1], [1, 1]];
PrintArray(A);
# [ [ 0, 1 ],
# [ 1, 1 ] ]
PrintArray(A^10);
# [ [ 34, 55 ],
# [ 55, 89 ] ]

[edit] Haskell

Instead of writing it directly, we can re-use the built-in exponentiation operator if we declare matrices as an instance of Num, using matrix multiplication (and addition). For simplicity, we use the inefficient representation as list of lists. Note that we don't check the dimensions (there are several ways to do that on the type-level, for example with phantom types).

import Data.List
 
xs <+> ys = zipWith (+) xs ys
xs <*> ys = sum $ zipWith (*) xs ys
 
newtype Mat a = Mat [[a]] deriving (Eq, Show)
 
instance Num a => Num (Mat a) where
negate (Mat x) = Mat $ map (map negate) x
Mat x + Mat y = Mat $ zipWith (<+>) x y
Mat x * Mat y = Mat [[xs <*> ys | ys <- transpose y] | xs <- x]
fromInteger _ = undefined -- don't know dimension of the desired matrix

Output:

*Main> Mat [[1,2],[0,1]]^4
Mat [[1,8],[0,1]]

This will work for matrices over any numeric type, including complex numbers. The implementation of (^) uses the fast binary algorithm for exponentiation.

Note: this implementation does not work for a power of 0.

[edit] J

mp=: +/ .*   NB. Matrix multiplication 
pow=: pow0=: 4 : 'mp&x^:y =i.#x'

or, from the J wiki, and faster for large exponents:

pow=: pow1=: 4 : 'mp/ mp~^:(I.|.#:y) x'

This implements an optimization where the exponent is represented in base 2, and repeated squaring is used to create a list of relevant powers of the base matrix, which are then combined using matrix multiplication. Note, however, that these two definitions treat a zero exponent differently (m pow0 0 gives an identity matrix whose shape matches m, while m pow1 0 gives a scalar 1).

Example use:

   (3 2,:2 1) pow 3
55 34
34 21

[edit] JavaScript

Works with: SpiderMonkey
for the print() and Array.forEach() functions.

Extends Matrix Transpose#JavaScript and Matrix multiplication#JavaScript

// IdentityMatrix is a "subclass" of Matrix
function IdentityMatrix(n) {
this.height = n;
this.width = n;
this.mtx = [];
for (var i = 0; i < n; i++) {
this.mtx[i] = [];
for (var j = 0; j < n; j++) {
this.mtx[i][j] = (i == j ? 1 : 0);
}
}
}
IdentityMatrix.prototype = Matrix.prototype;
 
// the Matrix exponentiation function
// returns a new matrix
Matrix.prototype.exp = function(n) {
var result = new IdentityMatrix(this.height);
for (var i = 1; i <= n; i++) {
result = result.mult(this);
}
return result;
}
 
var m = new Matrix([[3, 2], [2, 1]]);
[0,1,2,3,4,10].forEach(function(e){print(m.exp(e)); print()})

output

1,0
0,1

3,2
2,1

13,8
8,5

55,34
34,21

233,144
144,89

1346269,832040
832040,514229

[edit] Julia

Matrix exponentiation is implemented by the built-in ^ operator.

julia> [1 1 ; 1 0]^10
2x2 Array{Int64,2}:
89 55
55 34

[edit] Liberty BASIC

There is no native matrix capability. A set of functions is available at http://www.diga.me.uk/RCMatrixFuncs.bas implementing matrices of arbitrary dimension in a string format.

 
MatrixD$ ="3, 3, 0.86603, 0.50000, 0.00000, -0.50000, 0.86603, 0.00000, 0.00000, 0.00000, 1.00000"
 
 
print "Exponentiation of a matrix"
call DisplayMatrix MatrixD$
print " Raised to power 5 ="
MatrixE$ =MatrixToPower$( MatrixD$, 5)
call DisplayMatrix MatrixE$
print " Raised to power 9 ="
MatrixE$ =MatrixToPower$( MatrixD$, 9)
call DisplayMatrix MatrixE$
 


Exponentiation of a matrix
| 0.86603 0.50000 0.00000 |
| -0.50000 0.86603 0.00000 |
| 0.00000 0.00000 1.00000 |

Raised to power 5 =
| -0.86604 0.50002 0.00000 |
| -0.50002 -0.86604 0.00000 |
| 0.00000 0.00000 1.00000 |

Raised to power 9 =
| -0.00002 -1.00004 0.00000 |
| 1.00004 -0.00002 0.00000 |
| 0.00000 0.00000 1.00000 |



[edit] Lua

Matrix = {}
 
function Matrix.new( dim_y, dim_x )
assert( dim_y and dim_x )
 
local matrix = {}
local metatab = {}
setmetatable( matrix, metatab )
metatab.__add = Matrix.Add
metatab.__mul = Matrix.Mul
metatab.__pow = Matrix.Pow
 
matrix.dim_y = dim_y
matrix.dim_x = dim_x
 
matrix.data = {}
for i = 1, dim_y do
matrix.data[i] = {}
end
return matrix
end
 
function Matrix.Show( m )
for i = 1, m.dim_y do
for j = 1, m.dim_x do
io.write( tostring( m.data[i][j] ), " " )
end
io.write( "\n" )
end
end
 
function Matrix.Add( m, n )
assert( m.dim_x == n.dim_x and m.dim_y == n.dim_y )
 
local r = Matrix.new( m.dim_y, m.dim_x )
for i = 1, m.dim_y do
for j = 1, m.dim_x do
r.data[i][j] = m.data[i][j] + n.data[i][j]
end
end
return r
end
 
function Matrix.Mul( m, n )
assert( m.dim_x == n.dim_y )
 
local r = Matrix.new( m.dim_y, n.dim_x )
for i = 1, m.dim_y do
for j = 1, n.dim_x do
r.data[i][j] = 0
for k = 1, m.dim_x do
r.data[i][j] = r.data[i][j] + m.data[i][k] * n.data[k][j]
end
end
end
return r
end
 
function Matrix.Pow( m, p )
assert( m.dim_x == m.dim_y )
 
local r = Matrix.new( m.dim_y, m.dim_x )
 
if p == 0 then
for i = 1, m.dim_y do
for j = 1, m.dim_x do
if i == j then
r.data[i][j] = 1
else
r.data[i][j] = 0
end
end
end
elseif p == 1 then
for i = 1, m.dim_y do
for j = 1, m.dim_x do
r.data[i][j] = m.data[i][j]
end
end
else
r = m
for i = 2, p do
r = r * m
end
end
 
return r
end
 
 
m = Matrix.new( 2, 2 )
m.data = { { 1, 2 }, { 3, 4 } }
 
n = m^4;
 
Matrix.Show( n )

[edit] Maple

Maple handles matrix powers implicitly with the built-in exponentiation operator:

> M := <<1,2>|<3,4>>;
> M ^ 2;

\left[\begin{array}{cc}
 7 & 15 \\
 10 & 22
\end{array}\right]

If you want elementwise powers, you can use the elementwise ^~ operator:

> M := <<1,2>|<3,4>>;
> M ^~ 2;

\left[\begin{array}{cc}
 1 & 9 \\
 4 & 16
\end{array}\right]

[edit] Mathematica

In Mathematica there is an distinction between powering elements wise and as a matrix. So m^2 will give m with each element squared. To do matrix exponentation we use the function MatrixPower. It can handle all types of numbers for the power (integers, floats, rationals, complex) but also symbols for the power, and all types for the matrix (numbers, symbols et cetera), and will always keep the result exact if the matrix and the exponent is exact.

a = {{3, 2}, {4, 1}};
MatrixPower[a, 0]
MatrixPower[a, 1]
MatrixPower[a, -1]
MatrixPower[a, 4]
MatrixPower[a, 1/2]
MatrixPower[a, Pi]

gives back:


\left(
\begin{array}{cc}
 1 & 0 \\
 0 & 1
\end{array}
\right)


\left(
\begin{array}{cc}
 3 & 2 \\
 4 & 1
\end{array}
\right)


\left(
\begin{array}{cc}
 -\frac{1}{5} & \frac{2}{5} \\
 \frac{4}{5} & -\frac{3}{5}
\end{array}
\right)


\left(
\begin{array}{cc}
 417 & 208 \\
 416 & 209
\end{array}
\right)


\left(
\begin{array}{cc}
 \frac{2 \sqrt{5}}{3}+\frac{i}{3} & \frac{\sqrt{5}}{3}-\frac{i}{3} \\
 \frac{2 \sqrt{5}}{3}-\frac{2 i}{3} & \frac{\sqrt{5}}{3}+\frac{2 i}{3}
\end{array}
\right)


\left(
\begin{array}{cc}
 \frac{(-1)^{\pi }}{3}+2\frac{5^{\pi }}{3} & \frac{5^{\pi }}{3}-\frac{1}{3} (-1)^{\pi } \\
 2\frac{5^{\pi }}{3}-\frac{2}{3} (-1)^{\pi } & \frac{2 (-1)^{\pi }}{3}+\frac{5^{\pi }}{3}
\end{array}
\right)

Symbolic matrices like {{i,j},{k,l}} to the power m give general solutions for all possible i,j,k,l, and m:

MatrixPower[{{i, j}, {k, l}}, m] // Simplify

gives back (note that the simplification is not necessary for the evaluation, it just gives a shorter output):


\left(
\begin{array}{cc}
 \frac{2^{-m-1} \left(\left(\sqrt{i^2-2 i l+4 j k+l^2}-i+l\right) \left(-\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m+\left(\sqrt{i^2-2 i l+4 j k+l^2}+i-l\right)
   \left(\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m\right)}{\sqrt{i^2-2 i l+4 j k+l^2}} & \frac{j 2^{-m} \left(\left(\sqrt{i^2-2 i l+4 j
   k+l^2}+i+l\right)^m-\left(-\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m\right)}{\sqrt{i^2-2 i l+4 j k+l^2}} \\
 \frac{k 2^{-m} \left(\left(\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m-\left(-\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m\right)}{\sqrt{i^2-2 i l+4 j k+l^2}} &
   \frac{2^{-m-1} \left(\left(\sqrt{i^2-2 i l+4 j k+l^2}+i-l\right) \left(-\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m+\left(\sqrt{i^2-2 i l+4 j k+l^2}-i+l\right)
   \left(\sqrt{i^2-2 i l+4 j k+l^2}+i+l\right)^m\right)}{\sqrt{i^2-2 i l+4 j k+l^2}}
\end{array}
\right)

Final note: Do not confuse MatrixPower with MatrixExp; the former is for matrix exponentiation, and the latter for the matrix exponential (E^m).

[edit] MATLAB

For exponents in the form of A*A*A*A*...*A, A must be a square matrix:

function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA^(exponent);

Otherwise, to take the individual array elements to the power of an exponent (the matrix need not be square):

function [output] = matrixexponentiation(matrixA, exponent)
output = matrixA.^(exponent);

[edit] Maxima

a: matrix([3, 2],
[4, 1])$
 
a ^^ 4;
/* matrix([417, 208],
[416, 209]) */
 
a ^^ -1;
/* matrix([-1/5, 2/5],
[4/5, -3/5]) */

[edit] OCaml

We will use some auxiliary functions

(* identity matrix *)
let eye n =
let a = Array.make_matrix n n 0.0 in
for i=0 to n-1 do
a.(i).(i) <- 1.0
done;
(a)
;;
 
(* matrix dimensions *)
let dim a = Array.length a, Array.length a.(0);;
 
(* make matrix from list in row-major order *)
let matrix p q v =
if (List.length v) <> (p * q)
then failwith "bad dimensions"
else
let a = Array.make_matrix p q (List.hd v) in
let rec g i j = function
| [] -> a
| x::v ->
a.(i).(j) <- x;
if j+1 < q
then g i (j+1) v
else g (i+1) 0 v
in
g 0 0 v
;;
 
(* matrix product *)
let matmul a b =
let n, p = dim a
and q, r = dim b in
if p <> q then failwith "bad dimensions" else
let c = Array.make_matrix n r 0.0 in
for i=0 to n-1 do
for j=0 to r-1 do
for k=0 to p-1 do
c.(i).(j) <- c.(i).(j) +. a.(i).(k) *. b.(k).(j)
done
done
done;
(c)
;;
 
(* generic exponentiation, usual algorithm *)
let pow one mul a n =
let rec g p x = function
| 0 -> x
| i ->
g (mul p p) (if i mod 2 = 1 then mul p x else x) (i/2)
in
g a one n
;;
 
(* example with integers *)
pow 1 ( * ) 2 16;;
(* - : int = 65536 *)

Now matrix power is simply a special case of pow :

let matpow a n =
let p, q = dim a in
if p <> q then failwith "bad dimensions" else
pow (eye p) matmul a n;;
 
matpow (matrix 2 2 [ 1.0; 1.0; 1.0; 0.0 ]) 10;;
(* - : float array array = [|[|89.; 55.|]; [|55.; 34.|]|] *)
 
(* use as infix operator *)
let ( ^^ ) = matpow;;
 
[| [| 1.0; 1.0|]; [| 1.0; 0.0 |] |] ^^ 10;;
(* - : float array array = [|[|89.; 55.|]; [|55.; 34.|]|] *)

[edit] Octave

Of course GNU Octave handles matrix and operations on matrix "naturally".

M = [ 3, 2; 2, 1 ];
M^0
M^1
M^2
M^(-1)
M^0.5

Output:

ans =

   1   0
   0   1

ans =

   3   2
   2   1

ans =

   13    8
    8    5

ans =

  -1.0000   2.0000
   2.0000  -3.0000

ans =

   1.48931 + 0.13429i   0.92044 - 0.21729i
   0.92044 - 0.21729i   0.56886 + 0.35158i

(Of course this is not an implementation, but it can be used as reference for the results)

[edit] PARI/GP

M^n

[edit] Perl

use strict;
package SquareMatrix;
use Carp; # standard, "it's not my fault" module
 
use overload (
'""' => \&_string, # overload string operator so we can just print
'*' => \&_mult, # multiplication, needed for expo
'*=' => \&_mult, # ditto, explicitly defined to trigger copy
'**' => \&_expo, # overload exponentiation
'=' => \&_copy, # copy operator
);
 
sub make {
my $cls = shift;
my $n = @_;
for (@_) {
# verify each row given is the right length
confess "Bad data @$_: matrix must be square "
if @$_ != $n;
}
 
bless [ map [@$_], @_ ] # important: actually copy all the rows
}
 
sub identity {
my $self = shift;
my $n = @$self - 1;
my @rows = map [ (0) x $_, 1, (0) x ($n - $_) ], 0 .. $n;
bless \@rows
}
 
sub zero {
my $self = shift;
my $n = @$self;
bless [ map [ (0) x $n ], 1 .. $n ]
}
 
sub _string {
"[ ".join("\n " =>
map join(" " => map(sprintf("%12.6g", $_), @$_)), @{+shift}
)." ]\n";
}
 
sub _mult {
my ($a, $b) = @_;
my $x = $a->zero;
my @idx = (0 .. $#$x);
for my $j (@idx) {
my @col = map($a->[$_][$j], @idx);
for my $i (@idx) {
my $row = $b->[$i];
$x->[$i][$j] += $row->[$_] * $col[$_] for @idx;
}
}
$x
}
 
sub _expo {
my ($self, $n) = @_;
confess "matrix **: must be non-negative integer power"
unless $n >= 0 && $n == int($n);
 
my ($tmp, $out) = ($self, $self->identity);
do {
$out *= $tmp if $n & 1;
$tmp *= $tmp;
} while $n >>= 1;
 
$out
}
 
sub _copy { bless [ map [ @$_ ], @{+shift} ] }
 
# now use our matrix class
package main;
 
my $m = SquareMatrix->make(
[1, 2, 0],
[0, 3, 1],
[1, 0, 0] );
print "### Order $_\n", $m ** $_ for 0 .. 10;
 
$m = SquareMatrix->make(
[ 1.0001, 0, 0, 1 ],
[ 0, 1.001, 0, 0 ],
[ 0, 0, 1, 0.99998 ],
[ 1e-8, 0, 0, 1.0002 ]);
 
print "\n### Matrix is now\n", $m;
print "\n### Big power:\n", $m ** 100_000;
print "\n### Too big:\n", $m ** 1_000_000;
print "\n### WAY too big:\n", $m ** 1_000_000_000_000;
print "\n### But identity matrix can handle that\n",
$m->identity ** 1_000_000_000_000;

[edit] Perl 6

subset SqMat of Array where { .elems == all(.[]».elems) }
 
multi infix:<*>(SqMat $a, SqMat $b) {[
for ^$a -> $r {[
for ^$b[0] -> $c {
[+] ($a[$r][] Z* $b[].map: *[$c])
}
]}
]}
 
multi infix:<**> (SqMat $m, Int $n is copy where { $_ >= 0 }) {
my $tmp = $m;
my $out = [for ^$m -> $i { [ for ^$m -> $j { +($i == $j) } ] } ];
loop {
$out = $out * $tmp if $n +& 1;
last unless $n +>= 1;
$tmp = $tmp * $tmp;
}
 
$out;
}
 
multi show (SqMat $m) {
my $size = 1;
for ^$m X ^$m -> $i, $j { $size max= $m[$i][$j].Str.chars; }
say join "\n", $m».fmt("%{$size}s");
}
 
my @m = [1, 2, 0],
[0, 3, 1],
[1, 0, 0];
 
for 0 .. 10 -> $order {
say "### Order $order";
show @m ** $order;
}

Output:

### Order 0
1 0 0
0 1 0
0 0 1
### Order 1
1 2 0
0 3 1
1 0 0
### Order 2
1 8 2
1 9 3
1 2 0
### Order 3
 3 26  8
 4 29  9
 1  8  2
### Order 4
11 84 26
13 95 29
 3 26  8
### Order 5
 37 274  84
 42 311  95
 11  84  26
### Order 6
 121  896  274
 137 1017  311
  37  274   84
### Order 7
 395 2930  896
 448 3325 1017
 121  896  274
### Order 8
 1291  9580  2930
 1465 10871  3325
  395  2930   896
### Order 9
 4221 31322  9580
 4790 35543 10871
 1291  9580  2930
### Order 10
 13801 102408  31322
 15661 116209  35543
  4221  31322   9580

[edit] PicoLisp

Uses the 'matMul' function from Matrix multiplication#PicoLisp

(de matIdent (N)
(let L (need N (1) 0)
(mapcar '(() (copy (rot L))) L) ) )
 
(de matExp (Mat N)
(let M (matIdent (length Mat))
(do N
(setq M (matMul M Mat)) )
M ) )
 
(matExp '((3 2) (2 1)) 3)

Output:

-> ((55 34) (34 21))

[edit] Python

Using matrixMul from Matrix multiplication#Python

>>> from operator import mul
>>> def matrixMul(m1, m2):
return map(
lambda row:
map(
lambda *column:
sum(map(mul, row, column)),
*m2),
m1)
 
>>> def identity(size):
size = range(size)
return [[(i==j)*1 for i in size] for j in size]
 
>>> def matrixExp(m, pow):
assert pow>=0 and int(pow)==pow, "Only non-negative, integer powers allowed"
accumulator = identity(len(m))
for i in range(pow):
accumulator = matrixMul(accumulator, m)
return accumulator
 
>>> def printtable(data):
for row in data:
print ' '.join('%-5s' % ('%s' % cell) for cell in row)
 
 
>>> m = [[3,2], [2,1]]
>>> for i in range(5):
print '\n%i:' % i
printtable( matrixExp(m, i) )
 
 
 
0:
1 0
0 1
 
1:
3 2
2 1
 
2:
13 8
8 5
 
3:
55 34
34 21
 
4:
233 144
144 89
>>> printtable( matrixExp(m, 10) )
1346269 832040
832040 514229
>>>

[edit] R

Library: Biodem
library(Biodem)
m <- matrix(c(3,2,2,1), nrow=2)
mtx.exp(m, 0)
# [,1] [,2]
# [1,] 1 0
# [2,] 0 1
mtx.exp(m, 1)
# [,1] [,2]
# [1,] 3 2
# [2,] 2 1
mtx.exp(m, 2)
# [,1] [,2]
# [1,] 13 8
# [2,] 8 5
mtx.exp(m, 3)
# [,1] [,2]
# [1,] 55 34
# [2,] 34 21
mtx.exp(m, 10)
# [,1] [,2]
# [1,] 1346269 832040
# [2,] 832040 514229

Note that non-integer powers are not supported with this function.

[edit] Racket

 
#lang racket
(require math)
 
(define a (matrix ((3 2) (2 1))))
 
;; Using the builtin matrix exponentiation
(for ([i 11])
(printf "a^~a = ~s\n" i (matrix-expt a i)))
 
;; Output:
;; a^0 = (array #[#[1 0] #[0 1]])
;; a^1 = (array #[#[3 2] #[2 1]])
;; a^2 = (array #[#[13 8] #[8 5]])
;; a^3 = (array #[#[55 34] #[34 21]])
;; a^4 = (array #[#[233 144] #[144 89]])
;; a^5 = (array #[#[987 610] #[610 377]])
;; a^6 = (array #[#[4181 2584] #[2584 1597]])
;; a^7 = (array #[#[17711 10946] #[10946 6765]])
;; a^8 = (array #[#[75025 46368] #[46368 28657]])
;; a^9 = (array #[#[317811 196418] #[196418 121393]])
;; a^10 = (array #[#[1346269 832040] #[832040 514229]])
 
;; But it could be implemented manually, using matrix multiplication
(define (mpower M p)
(cond [(= p 1) M]
[(even? p) (mpower (matrix* M M) (/ p 2))]
[else (matrix* M (mpower M (sub1 p)))]))
(for ([i (in-range 1 11)])
(printf "a^~a = ~s\n" i (matrix-expt a i)))
 

[edit] Ruby

Ruby's standard library already provides the matrix-exponentiation operator. It is Matrix#** from package 'matrix' of the standard library. MRI 1.9.x implements the matrix-exponentiation operator in file matrix.rb, def ** (around line 961).

$ irb
irb(main):001:0> require 'matrix'
=> true
irb(main):002:0> m=Matrix[[3,2],[2,1]]
=> Matrix[[3, 2], [2, 1]]
irb(main):003:0> m**0
=> Matrix[[1, 0], [0, 1]]
irb(main):004:0> m ** 1
=> Matrix[[3, 2], [2, 1]]
irb(main):005:0> m ** 2
=> Matrix[[13, 8], [8, 5]]
irb(main):006:0> m ** 5
=> Matrix[[987, 610], [610, 377]]
irb(main):007:0> m ** 10
=> Matrix[[1346269, 832040], [832040, 514229]]

Starting with Ruby 1.9.3, it can also calculate Matrix ** Float.

Works with: Ruby version 1.9.3
irb(main):008:0> m ** 1.5
=> Matrix[[(6.308803769316981-0.03170173099577213i), (3.8990551577913446+0.05129
4478253365354i)], [(3.899055157791345+0.05129447825336536i), (2.4097486115256355
-0.0829962092491375i)]]

With older Ruby, it raises an exception for Matrix ** Float.

irb(main):008:0> m ** 1.5
ExceptionForMatrix::ErrOperationNotDefined: This operation(**) can't defined
        from /usr/lib/ruby/1.8/matrix.rb:665:in `**'
        from (irb):8

[edit] Scala

class Matrix[T](matrix:Array[Array[T]])(implicit n: Numeric[T], m: ClassManifest[T])
{
import n._
val rows=matrix.size
val cols=matrix(0).size
def row(i:Int)=matrix(i)
def col(i:Int)=matrix map (_(i))
 
def *(other: Matrix[T]):Matrix[T] = new Matrix(
Array.tabulate(rows, other.cols)((row, col) =>
(this.row(row), other.col(col)).zipped.map(_*_) reduceLeft (_+_)
))
 
def **(x: Int)=x match {
case 0 => createIdentityMatrix
case 1 => this
case 2 => this * this
case _ => List.fill(x)(this) reduceLeft (_*_)
}
 
def createIdentityMatrix=new Matrix(Array.tabulate(rows, cols)((row,col) =>
if (row == col) one else zero)
)
 
override def toString = matrix map (_.mkString("[", ", ", "]")) mkString "\n"
}
 
object MatrixTest {
def main(args:Array[String])={
val m=new Matrix[BigInt](Array(Array(3,2), Array(2,1)))
println("-- m --\n"+m)
 
Seq(0,1,2,3,4,10,20,50) foreach {x =>
println("-- m**"+x+" --")
println(m**x)
}
}
}
Output:
-- m --
[3, 2]
[2, 1]
-- m**0 --
[1, 0]
[0, 1]
-- m**1 --
[3, 2]
[2, 1]
-- m**2 --
[13, 8]
[8, 5]
-- m**3 --
[55, 34]
[34, 21]
-- m**4 --
[233, 144]
[144, 89]
-- m**10 --
[1346269, 832040]
[832040, 514229]
-- m**20 --
[2504730781961, 1548008755920]
[1548008755920, 956722026041]
-- m**50 --
[16130531424904581415797907386349, 9969216677189303386214405760200]
[9969216677189303386214405760200, 6161314747715278029583501626149]

[edit] Scheme

For simplicity, the matrix is represented as a list of lists, and no dimension checking occurs. This implementation does not work when the exponent is 0.

 
(define (dec x)
(- x 1))
 
(define (halve x)
(/ x 2))
 
(define (row*col row col)
(apply + (map * row col)))
 
(define (matrix-multiply m1 m2)
(map
(lambda (row)
(apply map (lambda col (row*col row col))
m2))
m1))
 
(define (matrix-exp mat exp)
(cond ((= exp 1) mat)
((even? exp) (square-matrix (matrix-exp mat (halve exp))))
(else (matrix-multiply mat (matrix-exp mat (dec exp))))))
 
(define (square-matrix mat)
(matrix-multiply mat mat))
 


Output:
> (matrix-exp '((3 2) (2 1)) 50)
((16130531424904581415797907386349 9969216677189303386214405760200)
 (9969216677189303386214405760200 6161314747715278029583501626149))

[edit] Seed7

The example below uses several features of Seed7:

  • Overloading of the operators * and ** .
  • The template enable_output, which allows writing a matrix with write (the function str must be defined before calling enable_output).
  • A for loop which loops over values listed in an array literal
$ include "seed7_05.s7i";
include "float.s7i";
 
const type: matrix is array array float;
 
const func string: str (in matrix: mat) is func
result
var string: stri is "";
local
var integer: row is 0;
var integer: column is 0;
begin
for row range 1 to length(mat) do
for column range 1 to length(mat[row]) do
stri &:= str(mat[row][column]);
if column < length(mat[row]) then
stri &:= ", ";
end if;
end for;
if row < length(mat) then
stri &:= "\n";
end if;
end for;
end func;
 
enable_output(matrix);
 
const func matrix: (in matrix: mat1) * (in matrix: mat2) is func
result
var matrix: product is matrix.value;
local
var integer: row is 0;
var integer: column is 0;
var integer: k is 0;
begin
product := length(mat1) times length(mat1) times 0.0;
for row range 1 to length(mat1) do
for column range 1 to length(mat1) do
product[row][column] := 0.0;
for k range 1 to length(mat1) do
product[row][column] +:= mat1[row][k] * mat2[k][column];
end for;
end for;
end for;
end func;
 
const func matrix: (in var matrix: base) ** (in var integer: exponent) is func
result
var matrix: power is matrix.value;
local
var integer: row is 0;
var integer: column is 0;
begin
if exponent < 0 then
raise NUMERIC_ERROR;
else
if odd(exponent) then
power := base;
else
# Create identity matrix
power := length(base) times length(base) times 0.0;
for row range 1 to length(base) do
for column range 1 to length(base) do
if row = column then
power[row][column] := 1.0;
end if;
end for;
end for;
end if;
exponent := exponent div 2;
while exponent > 0 do
base := base * base;
if odd(exponent) then
power := power * base;
end if;
exponent := exponent div 2;
end while;
end if;
end func;
 
const proc: main is func
local
var matrix: m is [] (
[] (4.0, 3.0),
[] (2.0, 1.0));
var integer: exponent is 0;
begin
for exponent range [] (0, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23) do
writeln("m ** " <& exponent <& " =");
writeln(m ** exponent);
end for;
end func;

Original source of matrix exponentiation: [1]

Output:

m ** 0 =
1.0, 0.0
0.0, 1.0
m ** 1 =
4.0, 3.0
2.0, 1.0
m ** 2 =
22.0, 15.0
10.0, 7.0
m ** 3 =
118.0, 81.0
54.0, 37.0
m ** 5 =
3406.0, 2337.0
1558.0, 1069.0
m ** 7 =
98302.0, 67449.0
44966.0, 30853.0
m ** 11 =
81883680.0, 56183720.0
37455816.0, 25699956.0
m ** 13 =
2363278336.0, 1621541248.0
1081027456.0, 741736960.0
m ** 17 =
1968565387264.0, 1350712688640.0
900475125760.0, 617852567552.0
m ** 19 =
56815568027648.0, 38983467794432.0
25988979228672.0, 17832093941760.0
m ** 23 =
47326274699395072.0, 32472478198530048.0
21648320946503680.0, 14853792205897728.0

[edit] Tcl

Using code at Matrix multiplication#Tcl and Matrix Transpose#Tcl

package require Tcl 8.5
namespace path {::tcl::mathop ::tcl::mathfunc}
 
proc matrix_exp {m pow} {
if { ! [string is int -strict $pow]} {
error "non-integer exponents not implemented"
}
if {$pow < 0} {
error "negative exponents not implemented"
}
lassign [size $m] rows cols
# assume square matrix
set temp [identity $rows]
for {set n 1} {$n <= $pow} {incr n} {
set temp [matrix_multiply $temp $m]
}
return $temp
}
 
proc identity {size} {
set i [lrepeat $size [lrepeat $size 0]]
for {set n 0} {$n < $size} {incr n} {lset i $n $n 1}
return $i
}
% print_matrix [matrix_exp {{3 2} {2 1}} 1]
3 2 
2 1 
% print_matrix [matrix_exp {{3 2} {2 1}} 0]
1 0 
0 1 
% print_matrix [matrix_exp {{3 2} {2 1}} 2]
13 8 
 8 5 
% print_matrix [matrix_exp {{3 2} {2 1}} 3]
55 34 
34 21 
% print_matrix [matrix_exp {{3 2} {2 1}} 4]
233 144 
144  89 
% print_matrix [matrix_exp {{3 2} {2 1}} 10]
1346269 832040 
 832040 514229 

[edit] TI-89 BASIC

This example is in need of improvement:
Explicitly implement exponentiation.

Built-in exponentiation:

[3,2;4,1]^4

Output: \begin{bmatrix}417 & 208 \\ 416 & 209\end{bmatrix}

[edit] Ursala

For matrices of floating point numbers, the library function mmult can be used as shown. The user-defined id function takes a square matrix to the identity matrix of the same dimensions. The mex function takes a pair (A,n) representing a real matrix A and a natural exponent n to the exponentiation An using the naive algorithm.

#import nat
#import lin
 
id = @h ^|CzyCK33/1.! 0.!*
mex = ||id@l mmult:-0^|DlS/~& iota

Alternatively, this version uses the fast binary algorithm.

mex = ~&ar^?\id@al (~&lr?/mmult@llPrX ~&r)^/~&alrhPX mmult@falrtPXPRiiX

This test program raises a 2 by 2 matrix to a selection of powers.

#cast %eLLL
 
test = mex/*<<3.,2.>,<2.,1.>> <0,1,2,3,4,10>

output:

<
   <
      <1.000000e+00,0.000000e+00>,
      <0.000000e+00,1.000000e+00>>,
   <
      <3.000000e+00,2.000000e+00>,
      <2.000000e+00,1.000000e+00>>,
   <
      <1.300000e+01,8.000000e+00>,
      <8.000000e+00,5.000000e+00>>,
   <
      <5.500000e+01,3.400000e+01>,
      <3.400000e+01,2.100000e+01>>,
   <
      <2.330000e+02,1.440000e+02>,
      <1.440000e+02,8.900000e+01>>,
   <
      <1.346269e+06,8.320400e+05>,
      <8.320400e+05,5.142290e+05>>>
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