0
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Is your Jacobian a symmetric matrix? $\endgroup$ – Carl Christian Dec 21 '17 at 10:39
  • $\begingroup$ @CarlChristian Yes it is $\endgroup$ – M a m a D Dec 21 '17 at 13:41
1
$\begingroup$

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".

$\endgroup$
  • $\begingroup$ Actually I am going to extend this problem to other kind of matrices. For example find a $Q$ such that $Q^T A Q$ equals to a matrix whose structure represents a special kind of graph. Can this be done? $\endgroup$ – M a m a D Dec 22 '17 at 7:05
  • $\begingroup$ @Drupalist I would say that the chances of the are remote. Tridiagonal or banded is as good as it gets in general. Perhaps some very special properties of a particular matrix might allow you to accomplish your goal. $\endgroup$ – Carl Christian Dec 25 '17 at 23:56

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.