How "chol" and "qz" MATLAB algorithms are utilised in the "eig" MATLAB function?
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The MATLAB function "eig" when used this way: [V,D] = eig(A,B), gives us the eigenvalues and eigenvectors of an eigenvalue problem A*V = B*V*D, where D is the diagonal matrix of eigenvalues and matrix V columns are the corresponding right eigenvectors.
In the documentation, it is menioned that the function "eig" uses the algorithms "chol" (Cholesky factorization), and "qz" (QZ factorization for generalized eigenvalues / generalized Schur decomposition).
However, when reading about the methods of Cholesky factorization and QZ factorization for generalized eigenvalues, the first method only decomposes a matrix into a product of where L is a lower triangular matrix, and the second method computes the eigenvalues and what about the eigenvectors?
So, for this reason, I don't understand how using only these algorithms could return us both eigenvectos and eigenvalues of an eigenvalue problem.
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Accepted Answer
Bruno Luong
on 13 Mar 2024
Edited: Bruno Luong
on 13 Mar 2024
6 Comments
Bruno Luong
on 14 Mar 2024
MATLAB documentation mentions only qz and chol because it has the user switch of eig. It doesn't mention other stage because it judges most users do not care about knowing such thing. ou can consider it as missing if you want.
I don't know exactly how MATLAB uses chol.
All I know is it can transform generalized eigen decomposition of sdp matrix (which has specific algorithm) with Cholesky decomposition:
% Generate random sdp matrices A and B
A=rand(5);
A=A'*A;
B=rand(5);
B=B'*B;
eig(A,B)
BB=chol(B); C=BB'\(A/BB);
C = (C + C')/2; % spd C
eig(C)
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