Is the Rayleigh quotient of a matrix invariant to a change of inner product?

Question

The Rayleigh quotient of a matrix is defined as

$$R_A(x) = \frac{(Ax, x)}{(x, x)},$$

and by the min-max theorem it can be used to obtain the eigenvalues of $A$. If we change the inner product does this then mean that the eigenvalues change?

Answer 1

Let $\mu $ be an eigenvalue of $A$ and let $x \ne 0$ sucht that $Ax= \mu x$. Then

$$ \frac{(Ax, x)}{(x, x)}= \frac{(\mu x, x)}{(x, x)}=\mu \frac{(x, x)}{(x, x)}= \mu.$$

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