Positive-definite Matrix - Simultaneous Diagonalization

Simultaneous Diagonalization

A symmetric, and a symmetric and positive-definite matrix can be simultaneously diagonalized, although not necessarily via a similarity transformation. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.

Let be a symmetric and a symmetric and positive-definite matrix. Write the generalized eigenvalue equation as where we impose that be normalized, i.e. . Now we use Cholesky decomposition to write the inverse of as . Multiplying by and, we get, which can be rewritten as where . Manipulation now yields where is a matrix having as columns the generalized eigenvectors and is a diagonal matrix with the generalized eigenvalues. Now premultiplication with gives the final result:

and, but note that this is no longer an orthogonal diagonalization.

Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other. For this result see Horn&Johnson, 1985, page 218 and following.