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Symbolic SVD for square matrix

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Diego Rodríguez
Diego Rodríguez on 17 May 2021
Commented: Diego Rodríguez on 18 May 2021
Hi everyone,
I'm developing a generic study of singular value decomposition (SVD) for square symetric matrix. The aim is to obtain generic functions where to substitute the generic coefficients of the matrix for functions to solve whether they became 0 or not. My code for the generic SVD computation is the following one:
clear;clc;
M = sym('M', [6 6]);
assume(M,'real');
M(2,1) = M(1,2);
M(3,1) = M(1,3);
M(3,2) = M(2,3);
M(4,1) = M(1,4);
M(4,2) = M(2,4);
M(4,3) = M(3,4);
M(5,1) = M(1,5);
M(5,2) = M(2,5);
M(5,3) = M(3,5);
M(5,4) = M(4,5);
M(6,1) = M(1,6);
M(6,2) = M(2,6);
M(6,3) = M(3,6);
M(6,4) = M(4,6);
M(6,5) = M(5,6);
disp(M);
detM = det(M);
disp('Determinant already computed... ');
svdM = svd(M);
The problem is that everything works fine when you describe a symbolic matrix with size smaller than 5 x 5. However, when I tested it with 6 x 6 or bigger matrixes, it seems that Matlab cannot handle the computation and never shows a results. I'll leave the code running for 5 days without obtaining any result, which is pretty strange because with 5x5 or below matrixes, it computes them perfectly under 2 minutes of time. Thus, I am wondering if this issue is some sort of limitation of symbolic computation of SVD? How could I compute efficiently the SVD for large symbolic matrixes?
Thanks in advance for any possible help!

Answers (1)

Christine Tobler
Christine Tobler on 17 May 2021
The computation of the SVD in symbolic mathematics involves the solving of a polynomial equation of degree given by the size of the input matrix. However, there is no closed-form solution in general for polynomial equations that are quintic or higher: https://en.wikipedia.org/wiki/Closed-form_expression#Example:_roots_of_polynomials
So the problem is not solvable.
  3 Comments
Diego Rodríguez
Diego Rodríguez on 18 May 2021
Thank you so much for your answers. I'll figure out another way to do what I'm aiming to. That's true, the complexity grows too quickly when working with symbolic matrixes. I'll close this issue since it has no clear solution.

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