MaplePrimes - answers and comments on Question, construct matrix of singular values

No way at all

Markiyan Hirnyk — Sun, 28 Aug 2011 09:50:51 Z

The singular values of a matrix do not determinate its entries. For example,
> with(LinearAlgebra);
> A := Matrix(2, 2, [[1, 2], [1, 2]]);
Matrix(2, 2, {(1, 1) = 1, (1, 2) = 2, (2, 1) = 1, (2, 2) = 2})
>SingularValues(A);
Vector(2, {(1) = 0, (2) = sqrt(10)})
>B := Matrix(2, 2, [[0, 0], [0, sqrt(10)]]);
>SingularValues(B);
Vector(2, {(1) = 0, (2) = sqrt(10)})

Edit. SingularValues(A,list); was replaced by SingularValues(A); .

Right and left singular vectors

Preben Alsholm — Sun, 28 Aug 2011 10:34:13 Z

In the help page for SingularValues you find:

"The singular values S together with the left singular Vectors as columns of Matrix U and the right singular Vectors as columns of Matrix Vt satisfy
(U . SS) . Vt = A
. Matrix SS has the entries of S along its main diagonal."

Example:
restart;
with(LinearAlgebra):
A := RandomMatrix(5,3,datatype=float);
S,U,Vt:=SingularValues(A,output=['S','U','Vt']);
S1:=DiagonalMatrix(S[1..3],5,3);
U.S1.Vt;
Equal(evalf(%),A);

DiagonalMatrix

pagan — Sun, 28 Aug 2011 11:01:16 Z

Maybe I read your question differently from others, but... are you just asking how to get S as a Matrix instead of a Vector?

# For A being an mxn Matrix, with S the Vector of singular values returned,
# then the DiagonalMatrix command can turn that S into a Matrix with the
# singular values along the main diagonal.

# eg.

LinearAlgebra:-DiagonalMatrix(S[1..min(m,n)],m,n);