LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Submatrix Determinants and Imaginary and Real Parts

Required data
Consider $$A\in \mathbb{S}^{n}$$, Let$$A_k \in\mathbb{S}^{k}$$ be a submatrix of A consisting of its first $$ k $$rows and columns, where $$k \leq n$$.

Matrix inequality
The matrix inequality $$ A>0 $$ is satisfied if and only if

$$det(A_k) > 0, k =1, ....., n$$.

Required data
Consider $$Q_R \in \mathbb{S}^{n}$$, $$Q_I \in \mathbb{R}^{n \times n}$$, $$ Q = Q^H = Q_R + j Q_I \in \mathbb{C}^{n \times n} $$,

Matrix inequality
The matrix inequality, $$Q>0$$ is equivalent to the matrix inequality given by

$$ \begin{bmatrix} Q_R & Q_I \\ -Q_I & Q_R \end{bmatrix}> 0$$.