Does maximizing the Rayleigh quotient using the method of Lagrange multipliers require the matrix to be positive semidefinite?

Question

I’m faced with the problem of maximizing a Rayleigh quotient:

$$\max_h \,\, \frac{h^t H h}{||h||^2}$$

Which is equivalent to solving

$$\max_h \,\, h^t H h $$ $$ s.t. ||h||^2=k >0, k \in \mathbb{R} $$

My matrix $H \in \mathbb{R}^{n \times n}$ is symmetric, but it is not positive semidefinite.

Now, the methods that I've seen for solving this sort of problem _assume_ that the matrix is symmetric and positive semidefinite (PSD). But, they seem to never use the PSD property anywhere. For example, consider the derivation given on the wikipedia page under the section "Special case of covariance matrices". Especially look at the section "Formulation using Lagrange multipliers". It seems to use the property that eigenvectors can be chosen to be orthogonal, but this is a property of symmetric matrices, and does not require PSDness.

So, I'm wondering - to maximize the Rayleigh quotient, must the matrix in the numerator be PSD? or is symmetric/Hermitian sufficient?

Related: Maximum of a generalized Rayleigh quotient

Answer 1

You do not need to assume $H$ is positive semidefinite.

Maximizing $x^THx$ (over the unit sphere) is equivalent to maximizing $x^T(H+cI)x$, and you can choose $c$ so $H+cI$ is positive semidefinite.