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.

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    $\begingroup$ 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

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".

  • $\begingroup$ 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

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