Given $T\in\mathrm M_n(\mathbb C)$, there exists $A$ and $B$ such that $A$ is Hermitian and $B$ is skew-Hermitian such that $T=A+B$.

Or, equivalently, $T=A+iB$ where $A$ and $B$ are both Hermitian matrices.

I wonder what is this decomposition called. My friend recalls it is the Toeplitz decomposition, but I barely find any information on that.

EDIT: The decomposition described above appears in many texts. To illustrate, here is an excerpt from page 145 of this article:

In [52], Kaluznin and Havidi approach the problem of unitary similarity for $m$-tuples, $(A_1,A_2,\ldots,A_n)$ and $(B_1,B_2,\ldots,B_n)$, of square matrices from a more geometric point of view. First note that when we decompose each matrix into its Hermitian and skew Hermitian components, $A_j = H_j + iK$, and $B_j = L_j + iM_j$, where $H_j$, $K_j$, $L_j$, and $M_j$ are all Hermitian, a unitary matrix $U$ satisfies $U^*A$, $U = B_j$ if and only if $U^*H_jU = L_j$ and $U^*K_jU = M_j$. Thus, we may replace each $A_j$ and $B_j$ with a pair of Hermitian matices and study the $2m$-tuples in which every entry is Hermitian.


1 Answer 1


I'm not saying there isn't a name, but it's certainly not universal.

I've been using this decomposition on an almost daily basis for decades, and I don't have a name for it. People usually just say that $A$ is the real part of $T$, and $B$ the imaginary part. Even in the case $n=1$ (i.e., the complex numbers).


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