Generalized spectral radius and matrix norm I am reading a paper that discusses the design of an approximate matrix in the context of numerical methods for PDEs. However, I do not understand the following step, which I believe should be independent of all the context:
We are trying to design a matrix $A$ such that its norm (when seen as an operator $A:V\rightarrow W$, where V and W are ordinary column vector spaces) is equal to that of another matrix $B$ that belongs to the same space. The inner product to be considered in $V$ is defined with the help of a positive-definite matrix $M$ as
$$
(v, w) = v^T M^{-1} w  \tag{1}
$$
where $v,w \in V$ and where its corresponding norm is denoted as $\left | \cdot \right |_{M^{-1}}$. The norm of any operator $C:V\rightarrow W$ is then expressed as
$$
\left \| C \right \| = \sup_{\left | x \right |_{M^{-1}} = 1}{x^T C x}  \label{norm} \tag{2}
$$
Closely paraphrasing the paper, a practical way to impose $\left \| A \right \| = \left \| B \right \|$ is to compute the spectrum of matrices $A$ and $B$, with respect to matrix $M^{-1}$ and impose that at least the spectral radius be the same. According to the explanations given in the same paper, the referred spectral radius (of an operator $C$) is to be taken as the largest of the absolute values of the solutions for $\lambda$ of the generalized eigenvalue problem $CU = \lambda M^{-1} U$.
Could someone help me break down the logic behind this equivalence? That is, why does the equality of the $M^{-1}$-spectral radii of $A$ and $B$ imply $\left \| A \right \| = \left \| B \right \|$?
EDIT:
I have realized that the matrices that the paper is applying this equivalence to are symmetric, so I am not sure (although I suspect so) if the equivalence is intended to be true only in this case, since it is not explicitly mentioned in the abstract discussion. For this case, I have found an explanation which I will post as an answer.
 A: I have found a way to explain this equivalence for the case of both $A$ and $B$ symmetric, based on this material.
Note that the norm above can be expressed as
$$
\left \| C \right \| = \sup_{x \neq 0}{R(x)}  \tag{2}
$$
where the generalized Rayleigh coefficient, $R(x)$, is defined as
$$
R(x) := \frac{x^T C x}{x^T M^{-1} x}
$$
The maximum of $R$ is attained at a stationary point of R (sum is assumed over repeated indices):
$$
\frac{\partial }{\partial x_m} \left ( \frac{x_i C_{ij} x_j}{x_k M^{-1}_{kl} x_l} \right ) = 0 \Rightarrow \left ( \delta_{im} C_{ij} x_j +  x_i C_{ij} \delta_{jm}\right ) x_k M^{-1}_{kl} x_l - x_i C_{ij} x_j \left ( \delta_{km}  M^{-1}_{kl} x_l - x_k M^{-1}_{kl} \delta_{lm}\right ) \\
= 2C_{jm} x_k M^{-1}_{kl} x_l - 2 x_i C_{ij} x_j M^{-1}_{ml} x_l = 0 \Rightarrow (x^T M^{-1} x) C x = (x^T C x) M^{-1}x
$$
where $\delta_{ij}$ is the Kronecker delta. Note that we have used the symmetry of $A$ and $B$. That is,
$$
C x = R(x) M^{-1}x
$$
So that the maximum is attained at an eigenvalue of the generalized problem. In particular, the maximum eigenvalue will yield the greatest value of $R$. Thus, imposing that the (generalized) spectral radius of the two matrices coincides will force their norms to be equal too.
