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.

 A: 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).
A: On Wikipedia, this fact appears on the page for Hermitian matrices. They cite: Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis, second edition. Cambridge University Press. ISBN 9780521839402. (on page 227) as saying that this is called the Toeplitz decomposition. However as another answer said, the name may not be universal, or even commonly known.
A: It's called the Toeplitz Decompostion as stated in a previous answer
Additionally, here's a quotation from Matrix Analysis (2nd Edition) by Roger A. Horn & Charles R. Johnson

Theorem (Toeplitz decomposition): Each $A ∈ Mn$ can be written uniquely as $A = H + i K$ , in which both H and K are Hermitian. It can also be written uniquely as $A = H + S$, in which H is Hermitian and S is skew-Hermitian.


Proof: Write $A = \frac{1}{2} (A + A^∗) + i[\frac{1}{2i} (A − A^∗)]$, and observe that both $H = \frac{1}{2} (A + A^∗)$ and $K = \frac{1}{2i} (A − A^∗)$ are Hermitian. For the uniqueness assertion, observe that if $A = E + iF$ with both $E$ and $F$ Hermitian, then $2H = A + A^∗ = (E + iF) + (E + iF)^∗ = E + iF + E^∗ − iF^∗ = 2E$, so $E = H$. Similarly, one shows that $F = K$. The assertions about the representation $A = H + S$ are proved in the same way.

