Find the singular value decomposition (SVD) of a matrix

Problem: Find the SVD for the matrix \(\begin {bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end {bmatrix} \) Notice that in Maple, the singular values matrix, normally called S, is returned as a column vector. So need to call DiagonalMatrix() to format it as expected.

Mathematica

mat = {{1,2,3}, 
       {4,5,6}} 
 
{u,s,v}=N[SingularValueDecomposition[mat]]

{{0.386318,-0.922366}, 
{0.922366,0.386318}} 
 
{{9.50803,0.,0.}, 
{0.,0.77287,0.}} 
 
{{0.428667,0.805964,0.408248}, 
{0.566307,0.112382,-0.816497}, 
{0.703947,-0.581199,0.408248}}

(*Reconstruct A from its components *) 
u.s.Transpose[v]

{{1.,2.,3.}, 
{4.,5.,6.}}

Matlab

A=[1 2 3; 
   4 5 6]; 
[u,s,v]=svd(A); 
u 
s 
v

u = 
   -0.3863   -0.9224 
   -0.9224    0.3863 
 
s = 
 
    9.5080         0         0 
         0    0.7729         0 
 
v = 
 
   -0.4287    0.8060    0.4082 
   -0.5663    0.1124   -0.8165 
   -0.7039   -0.5812    0.4082

u*s*v'

ans = 
    1.0000    2.0000    3.0000 
    4.0000    5.0000    6.0000

Maple

restart; 
with(LinearAlgebra): 
A:=Matrix([[1,2,3],[4,5,6.]]);

     [1    2    3 ] 
A := [            ] 
     [4    5    6.]

m,n:=Dimensions(A): 
u,s,v:=SingularValues(A, 
    output=['U', 'S', 'Vt']): 
u;

 [0.386317703118612    -0.922365780077058] 
 [                                       ] 
 [0.922365780077058    0.386317703118612 ]

s:=DiagonalMatrix(s,m,n); 
v;

     [9.50803200069572            0            0] 
s := [                                          ] 
     [       0            0.772869635673485    0] 
 
     [0.428667133548626    0.566306918848035     0.703946704147444 ] 
v:=  [0.805963908589298    0.112382414096594     -0.581199080396110] 
     [0.408248290463863    -0.816496580927726    0.408248290463863 ]

> s2:=DiagonalMatrix(SingularValues(A, 
                     output=['S']),m,n);

      [9.50803200069572        0            0] 
s2 := [                                      ] 
      [       0         0.772869635673485    0]

> evalf(u.s.v);

[1.    2.    3.] 
[              ] 
[4.    5.    6.]

2.35 Find the singular value decomposition (SVD) of a matrix