Singular value decomposition

$A$ is any m by n matrix, square or rectangular. Its rank is r. We will diagonalize this A, but not by Failed to parse (syntax error): {\displaystyle X^{−1}AX} . The eigenvectors in $X$ have three big problems: They are usually not orthogonal, there are not always enough eigenvectors, and $Ax$ = Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle λx} requires $A$ to be a square matrix. The singular vectors of $A$ solve all those problems in a perfect way.

The Singular Value Decomposition (SVD)

According to the web page above, for any rectangular matrix $A$ , we can decomposite it as Failed to parse (syntax error): {\displaystyle A=UΣV^T}

Task Description

Firstly, input two numbers "m" and "n".

Then, input a square/rectangular matrix $A^{m\times n}$ .

Finally, output Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U,Σ,V} with respect to $A$ .

Example

Sample Input

2 2
3 0
4 5

From the input above we can know that $A$ is a 2 by 2 matrix.

Sample Output

   
0.31622776601683794 -0.9486832980505138
0.9486832980505138 0.31622776601683794

6.708203932499369 0
0 2.23606797749979

0.7071067811865475 -0.7071067811865475
0.7071067811865475 0.7071067811865475

The output may vary depending your choice of the data types.

Julia

Julia has an svd() function as part of its built-in LinearAlgebra package.

julia> using LinearAlgebra

julia> function testsvd()
           rows, cols = [parse(Int, s) for s in split(readline())]
           arr = zeros(rows, cols)
           for row in 1:rows
               arr[row, :] .= [tryparse(Float64, s) for s in split(readline())]
           end
           display(svd(arr))
       end
testsvd (generic function with 1 method)

julia> testsvd()
2 2
3 0
4 5
SVD{Float64, Float64, Matrix{Float64}, Vector{Float64}}
U factor:
2×2 Matrix{Float64}:
 -0.316228  -0.948683
 -0.948683   0.316228
singular values:
2-element Vector{Float64}:
 6.70820393249937
 2.2360679774997894
Vt factor:
2×2 Matrix{Float64}:
 -0.707107  -0.707107
 -0.707107   0.707107

Python

This implementation uses "numpy" library.

from numpy import *
A = matrix([[3, 0], [4, 5]])
U, Sigma, VT = linalg.svd(A)
print(U)
print(Sigma)
print(VT)

Output:

[[-0.31622777 -0.9486833 ]
 [-0.9486833   0.31622777]]
[6.70820393 2.23606798]
[[-0.70710678 -0.70710678]
 [-0.70710678  0.70710678]]