MATLAB: Cholesky of a matrix with deleted rows and columns: chol(QAQ’)

choleskymatrix

For a large matrix A, Cholesky factor L exists

L = chol(A)

B is a matrix of smaller dimension obtained by deleting rows and columns via a 0/1 filter matrix:

B = Q*A*Q'

Is there a way of computing chol(B) or chol(Q*A*Q') directly from A, Q and L, without actually computing B?

Thanks

Best Answer

 [~,L_new] = qr(L*Q.',0);

Related Solutions

MATLAB: I have a matrix q and i need the diagonal of qq’

In your Help Navigator, search for "arithmetic operators" and "diag"

diag(q*q')

is the same as

diag(q*transpose(q))

if all of the elements in q are real. Otherwise ' will give you the complex-conjugate transpose.

MATLAB: How to compute only the matrix Q in QR factorization in MATLAB 7.10 (R2010a)

In QR factorization of an m-by-n matrix A where n > m, only the first m columns of A are needed to compute Q. Therefore, you can obtain Q from the smaller m-by-m matrix formed from the first m columns of A. The following small example demonstrates this idea:

m = 40;
n = 100;
A = rand(m,n);
[Q, ~] = qr(A);
Asmall = A(:,1:m);  % left m-by-m block from A
[Q2, ~] = qr(Asmall);
% check that Q, Q2 are equal to within roundoff error
err = norm(Q - Q2)

Note that this approach will work even if A is full rank but 'Asmall' is not full rank. In general, the QR decomposition is not unique. Even in the case of 'Asmall' not being full rank, Q will still be unitary, and there will still exist an R such that Q*R = A for the given Q matrix. The first m columns of such an R are computed (although we are not interested in them). The m+1:n columns of R will not be computed, but there is no doubt about their existence. They will be whatever they need to be so that Q*R=A.

Best Answer

Related Solutions

MATLAB: I have a matrix q and i need the diagonal of qq’

MATLAB: How to compute only the matrix Q in QR factorization in MATLAB 7.10 (R2010a)

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