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This seems simple, but I can't seem to solve it. Let $$A = \begin{pmatrix} D_{11} & \dots & D_{1n} \\ D_{21} & \dots & D_{2n} \\ \vdots & \vdots & \vdots\\ D_{n1} & \dots & D_{nn} \end{pmatrix}$$ be a block matrix such that (1) $A$ is symmetric (2) each $D_{ij}$ is diagonal. The goal is to find the eigenspectrum of $A$.

Special case) If $D_{ij} = c_{ij}I$, would replacing each block $D_{ij}$ by the scalar $c_{ij}$ give the same eigenspectrum?

General) Without this additional assumption, can we compute the eigenspectrum of $A$ using a surrogate matrix $\tilde{A}$ with block entries $A_{ij}$ replaced by a characterization of $D_{ij}$? If so, what would be this characterization?

I have seen this post where we can reorder the rows/columns to preserve eigenspectrum and reduce computation, but I think this question is a bit more specific and hopefully there exists a more analytical answer.

Rodrigo de Azevedo
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rhdxor
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2 Answers2

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For the general case, suppose that the diagonal matrices have size $m \times m$, and let $d_{ijk}$ denote the $k$th diagonal entry of the diagonal matrix $D_{ij}$. Your matrix can be written in the form $$ A = \sum_{i,j = 1}^n \sum_{k = 1}^m d_{ijk} \cdot E_{ij}^{(n)} \otimes E_{kk}^{(m)} $$ where $\otimes$ denotes a Kronecker product and $E_{ij}^{(n)}$ is the $n \times n$ matrix with a $1$ as its $i,j$ entry and zeros elsewhere. This matrix is similar to the matrix $$ B = \sum_{i,j = 1}^n \sum_{k = 1}^m d_{ijk} \cdot E_{kk}^{(m)} \otimes E_{ij}^{(n)}, $$ which has the block-diagonal form $$ B = \pmatrix{B_1\\ & \ddots \\ && B_n} $$ where the $i,j$ entry of $B_k$ is $d_{ijk}$.

In other words, the spectrum of $A$ is the combined spectrum of each of the $n \times n$ matrices $B_1,\dots,B_m$.

Ben Grossmann
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In the special case where $D_{ij} = c_{ij}I_d$, there is an answer using Kronecker product, because you can write your matrix:

$$A=c \otimes I_d$$

As a consequence, the eigenvalues of $A$ are all the possible products of eigenvalues of $c$ with eigenvalues of $I_d$; otherwise said, the spectrum of $D$ is the spectrum of $c$, each eigenvalue having multiplicity $d$.

Jean Marie
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