I have the following problem:
I need to find the fastest way to calculate the eigenvalues of a matrix that is the product of two Toeplitz matrices. $B = A U$.
The first is a regular Toeplitz matrix $A$, while the second is an upper triangular matrix $U$.
I thought to use the QR decomposition on $A$ in such a way that $A U = (Q R) U = Q (R U)$
Since the product of two upper triangular matrices is an upper triangular matrix I would still get a QR type factorisation. Now I have two questions:
- Is this the best way to calculate eigenvalues or are there more efficient ways since they are structured matrices?
- If the proposed way is correct, which is the best algorithm to factorize the Toeplitz A matrix? Is it better to transform the Toeplitz matrix into a Hessenberg matrix?
Thank you very much