Singular value decomposition
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 have three big problems: They are usually not orthogonal, there are not always enough eigenvectors, and = Failed to parse (syntax error): {\displaystyle λx} requires to be a square matrix. The singular vectors of solve all those problems in a perfect way.
You are encouraged to solve this task according to the task description, using any language you may know.
The Singular Value Decomposition (SVD)
According to the web page above, for any rectangular matrix , 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 .
Finally, output Failed to parse (syntax error): {\displaystyle U,Σ,V} with respect to .
Example
Sample Input
2 2 3 0 4 5
From the input above we can know that 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.
Remark
It’s encouraged to implement the algorithm by yourself while using libraries is still acceptible.
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
Phix
with javascript_semantics requires("1.0.2") -- builtins/svd.e added, haha include builtins/svd.e sequence a = {{3,0}, {4,5}} sequence {u,w,v} = svd(a) ?u ?w ?v
- Output:
{{0.9486832981,0.316227766},{-0.316227766,0.9486832981}} {2.236067977,6.708203932} {{0.7071067812,0.7071067812},{-0.7071067812,0.7071067812}}
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]]