Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

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):

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:

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 $$\frac{1}{\hat{N}}$$.

• 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 '18 at 7:19