Say $M := \begin{pmatrix} A & B\\ C & D \end{pmatrix}$ is a block matrix with $A, D$ being square matrices and this $B$ and $C^T$ having the same shape. Is there any norm characterizing the collection of $B$ and $C$ that is invariant under all block diagonal similarity transformations $M\to S M S^{-1}$ with $S=\begin{pmatrix}E & 0\\ 0 & F\end{pmatrix}$ of the whole block matrix? Were $B$ and $C$ square matrices I'd use e.g. the sum of squared eigenvalues, but is there something similar for non-square matrices?
In brief, the requirements for the block norms I seek are:
- the norm of the square blocks $A$ and $D$ must be invariant (under these similarity transformations of $M$) respectively
- either there are individual norms for $B$ and $C$ which are invariant, or the is one combined norm depending on $B$ and $C$ which is invariant
Is the latter requirement possible?
Due to the block diagonal structure of $S$, all blocks transform independently, so there should be independent norms for $B$ and $C$. But since $B\to F B E^{-1}$ and $C\to E C F^{-1}$ are no similarity transformations, I couldn't even use eigenvalues if $B$ and $C$ were square matrices. However, $BC \to F BC F^{-1}$ and $CB \to E CB E^{-1}$ are similarity transformations, the remaining question is what kind of norm to use and maybe also how to decide whether $BC$ or $CB$ or both are "significant"…
Best Answer
Let $X$ be an $m \times n$ matrix. A norm $\|X\|$ is called unitarily invariant if $\|UXV\| = \|X\|$ for appropriately shaped unitary matrices $U$ and $V$ (i.e., $U^*U=I_m$ and $V^*V=I_n$.
Now, the singular value decomposition of $X$ tells us that if $\|\cdot\|$ is a unitarily invariant norm, it will be a function only of the singular values of $X$ (a so-called spectral function).
A simple observation is that if $X$ is square and diagonalizable, then we can select $S=V$, the matrix of eigenvectors of $X$, and consequently the norms that you are looking for again must be unitarily (here orthogonally) invariant.
Please have a look in the book Matrix Analysis by Horn and Johnson, who devote an entire chapter (or more) to matrix norms.