# Every complex matrix can be written as the sum of a Hermitian and a skew-Hermitian matrix. What is this called?

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.

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).