A

is any m by n matrix, square or rectangular. Its rank is r. We will diagonalize this A, but not by

X^{−1}AX

. The eigenvectors in

X

have three big problems: They are usually not orthogonal, there are not always enough eigenvectors, and

Ax

λx

requires

A

to be a square matrix. The singular vectors of

A

solve all those problems in a perfect way. [https://math.mit.edu/classes/18.095/2016IAP/lec2/SVD_Notes.pdf The Singular Value Decomposition (SVD)] According to the web page above, for any rectangular matrix

A

, we can decomposite it as

A=UΣV^T

''' Task Description''' Firstly, input two numbers "m" and "n". Then, input a square/rectangular matrix

A^{m\times n}

. Finally, output

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. '''Remark''' 1. It’s encouraged to implement the algorithm by yourself while using libraries is still acceptible. 2. The algorithm should be applicable for general case(

m\times n