LMIs in Control/Linear matrix inequalities and control theory/pages/Fundamentals of Matrix and LMIs/Fundamentals of Matrix and LMI/Basic Matrix Theory

Basic Matrix Notation
Consider the complex matrix $$A\in \C^{n\times m} $$.
 * $$ A=\begin{bmatrix} a_{11} & \dots & a_{1m} \\

\vdots & \ddots & \vdots \\ a_{n1} & \dots & a_{nm}\end{bmatrix} \in \C^{n\times m} $$

Transpose of a Matrix

The transpose of $$A$$, denoted as $$A^{T}$$ or $$A'$$ is:
 * $$ A^T=\begin{bmatrix} a_{11} & \dots & a_{n1} \\

\vdots & \ddots & \vdots \\ a_{1m} & \dots & a_{nm}\end{bmatrix} \in \C^{m\times n}. $$

Adjoint of a Matrix

The adjoint or hermitian conjugate of $$A$$, denoted as $$A^{*}$$ is:
 * $$ A^*=\begin{bmatrix} a_{11}^* & \dots & a_{n1} ^*\\

\vdots & \ddots & \vdots \\ a_{1m}^* & \dots & a_{nm}^*\end{bmatrix} \in \C^{m\times n}. $$ Where $$a_{nm}^*$$ is the complex conjugate of matrix element $$a_{nm}$$.

Notice that for a real matrix $$A\in\R^{n\times m}$$, $$A^*=A^T$$.

Important Properties of Matricies
Hermitian, Self-Adjoint, and Symmetric Matricies

A square matrix $$A\in \C^{n\times n} $$ is called Hermitian or self-adjoint if $$A=A^*$$.

If $$A\in \R^{n\times n} $$ is Hermitian then it is called symmetric.

Unitary Matricies

A square matrix $$A\in \C^{n\times n} $$ is called unitary if $$A^*=A^{-1}$$ or $$A^*A=I$$.