We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller than 1. Here $\otimes$ is tensor product. These matrices $\bf J$, $\hat{\bf{G}}$, $\hat{\bf{H}}$ are defined below. I am thinking about if there is a matrix norm $\|\|$ such that $\|\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}\| < 1$, because spectral radius is smaller than any matrix norm.

  1. $\bf {J}$ is a general real-valued square matrix. We can assume the the maximum absolute value in $\bf {J}$ is smaller than some threshold $\epsilon > 0$, but it is NOT allowed to assume $\bf J$ is some special type of matrix such as "symmetric matrix", "positive-definite matrix" or "non-negative matrix". $\bf I$ is an identity matrix of the same size as $\bf J$.

  2. Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (sorry I missed the coefficient before):

    enter image description here

  3. Let $\hat{\mathbf{G}}=\begin{pmatrix} {\bf G} & & & & \\ & \ddots & & & \\ & & {\bf G} & & \\ & & & \ddots & \\ & & & & {\bf G} \\ \end{pmatrix}$ be a $p\hat{N}\times p\hat{N}$ diagonal block matrix repeating $p$ times of $\bf G$, which is an $\hat{N} \times \hat{N}$ matrix defined as the following:

    enter image description here where $h$ is some positive coefficient. The eigenvalues of ${\bf G}$ has analytical form $\frac{h}{{{\rm{2}}(\cos (\frac{{k\pi }}{{\widehat N + 1}}) - 1)}}$ where $k = 1,...,\hat{N}$.

  4. Let $\hat{\mathbf{W}}=\begin{pmatrix} {\bf W} & & & & \\ & \ddots & & & \\ & & {\bf W} & & \\ & & & \ddots & \\ & & & & {\bf W} \\ \end{pmatrix}$ be a $p\hat{N}\times p\hat{N}$ diagonal weight matrix repeating $p$ times of $\bf {W}$, which is a diagonal weight matrix where all diagonal elements are non-negative and sum to $1$. The weights are free to choose, therefore we can simple choices like every element in $\bf W$ is $\frac{1}{\hat{N}}$.

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  • Actually, both questions, that about the spectral radius and that about a matrix norm, are equivalent. This is stated in Householder's Theorem: the spectral radius of $A$ equals the infimum of all matrix (subordinated) norms $\|A\|$. – Denis Serre Dec 8 at 7:19

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