Finding a special orthogonal matrix to generate pentadiagonal matrix

Question

Suppose a Jacobian matrix $A_{n \times n}$ is given. I need to find an orthogonal matrix $Q_{n \times n}$ such that $Q^T A Q = B$ and $B$ is a pentadiagonal matrix. I need to know if this problem can be solved and is there a solution to generalize this problem to other kind of matrices? The Jacobian matrix is symmetric.

Answer 1

Any matrix square matrix $A$ can be reduced to upper Hessenberg form $H = Q^T A Q$ using orthogonal similarity transformations. If $A$ is symmetric, then $H$ is symmetric, hence tridiagonal. In general, the reduction to Hessenberg form is a standard preprocessing step before applying the QR algorithm for computing eigenvalues. The basic algorithm is thoroughly described in Golub and van Loan's book "Matrix Computations".