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1. Motivation

Consider symmetric matrices $A,B\in\mathbb{R}^{n\times n}$, and let $A$ be full-rank and $B$ be low-rank. The simultaneous block-diagonalization, defined as the following

$$A=V_{1}\hat{A}_{1}V_{1}^{T}+V_{2}\hat{A}_{2}V_{2}^{T},\qquad B=V_{1}\hat{B}_{1}V_{1}^{T},$$

is extremely useful because it simplifies the parabolic equation $\frac{d}{dt}x=Ax+Bu$ into two distinct and decoupled systems,

$$\frac{d}{dt}\left(V_{1}^{T}x\right) =\hat{A}_{1}\left(V_{1}^{T}x\right)+\hat{B}_{1}\left(V_{1}^{T}u\right),\\ \frac{d}{dt}\left(V_{2}^{T}x\right) =\hat{A}_{2}\left(V_{2}^{T}x\right).$$

Assuming commutivity, $AB=BA$, choosing $V_{1},V_{2}$ is a simple matter of partitioning the shared eigenspace of $A,B$. However, when A,B are non-commutative, their eigenspaces may be very different, and a block-diagonalization cannot be exactly achieved. We desire an approximation, in which the "strength" of the off-diagonal blocks is bounded with respect to the dominant diagonal blocks.

$$\|V_{1}^{T}AV_{2}\|_{F}\le\alpha\|V_{1}^{T}AV_{1}\|_{F},\qquad\|V_{1}^{T}BV_{2}\|_{F}\le\alpha\|V_{1}^{T}BV_{1}\|_{F}.$$

Hence we are motivated by the parabolic PDE context to pose the following question.

In the non-commutative case, i.e. $AB\neq BA$, how does one chose $V_{1},V_{2}$ to approximate a simultaneous block-diagonalization without setting $V_{2}=0$?

2. View via representation theory.

The same question can be posed in a more general manner using the language of representation theory. (Our original problem is recovered by writing $A_2:=V_2 V_2^T A V_2 V_2^T$ and $A_1:=A-A_2$.)

Given symmetric, non commutative matrices, $A,B$, is there a decomposition, $A=A_{1}+A_{2},$ such that $A_{2}\neq0$ is (approximately) commutative with $A_{1}$ and $B$, and that the rank of $A_{1}$ is minimized?

In other words, can we find an $A_{2}$ such that $A_{1}A_{2}\approx A_{2}A_{1}$ and $BA_{2}\approx A_{2}B$, while allowing for $BA_{1}\neq A_{1}B$?

In the commutative matrix context, it may be easier to redefine "approximate" w.r.t. the norm of the commutator,

$$AB\approx BA\text{ iff }\|AB-BA\|_{F}\le\alpha\|AB\|_{F}.$$

There is strong reason to believe that low-rank has an important role in this context. Choosing $A_2$ by projecting $A$ onto the kernel / nullspace of $B$, we are guaranteed to satisfy $A_2 B = B A_2=0$. However, it is unlikely for this to simultaneously satisfy $A_{1}A_{2}\approx A_{2}A_{1}$. Likewise, we can guarantee that $A_{1}A_{2} = A_{2}A_{1} =0$ by partitioning the eigenspace of $A$. However, in this case, there is no guarantee that $BA_{2}\approx A_{2}B$.

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  • $\begingroup$ Presumably you don't want $A_2$ to be a scalar multiple of the identity matrix. What other conditions do you have? $\endgroup$ Nov 7, 2014 at 17:38
  • $\begingroup$ @RobertIsrael: Thanks for the hint -- I will add to description. Of all viable solutions, we wish to minimize the dimensionality of $A_1$, i.e. its rank. $\endgroup$ Nov 7, 2014 at 18:23
  • $\begingroup$ How "approximate" do you want the approximation to be? $\endgroup$ Nov 7, 2014 at 18:26
  • $\begingroup$ @RobertIsrael: Ideally, the approximation could be bounded in a norm, relative to the original matrices. E.g. $\|A_1 A_2 - A_2 A_1\|_F \le \alpha \|A_1 A_2\|_F $. $\endgroup$ Nov 7, 2014 at 18:33

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