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

Submatrix Determinants

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

$det(A_{k})>0,k=1,.....,n$.

Imaginary and Real Parts

Required data

Consider $Q_{R}\in \mathbb {S} ^{n}$, $Q_{I}\in \mathbb {R} ^{n\times n}$, $Q=Q^{H}=Q_{R}+jQ_{I}\in \mathbb {C} ^{n\times n}$,

Matrix inequality

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

${\begin{bmatrix}Q_{R}&Q_{I}\\-Q_{I}&Q_{R}\end{bmatrix}}>0$.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.