Notation: what is He(A) and Sh(A)?

Question

In the book "Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach" in "Conventions and Notation" section there are two expressions with "fairly standard notation": $$\text{He} A = \frac{A + A^{*}}{2} \quad \text{and} \quad \text{Sh} A = \frac{A - A^{*}}{2},$$ where $A$ is a $m \times n$ matrix.

Why $\text{He}$ and $\text{Sh}$? Do they have some special names and properties?
$\text{Sh}A$ looks like hyperbolic sine in exponential form, but I can't see the connection between these two...

Answer 1

It stands for the Hermitian part and the skew-Hermitian part of the matrix $A$.

See http://mathworld.wolfram.com/HermitianPart.html, for example.