Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
Linear Matrix Inequalities in Control
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Recap: Hermitian and symmetric matrices<br />
Def<strong>in</strong>ition<br />
For a real or complex matrix A the <strong>in</strong>equality A ≺ 0 means that A is<br />
Hermitian and negative def<strong>in</strong>ite.<br />
A is Hermitian is A = A ∗ = Ā⊤ . If A is real this amounts to A = A ⊤<br />
and we call A symmetric.<br />
Set of n × n Hermitian or symmetric matrices: H n and S n .<br />
All eigenvalues of Hermitian matrices are real.<br />
By def<strong>in</strong>ition a Hermitian matrix A is negative def<strong>in</strong>ite if<br />
u ∗ Au < 0 for all complex vectors u = 0<br />
A is negative def<strong>in</strong>ite if and only if all its eigenvalues are negative.<br />
A B, A ≻ B and A B def<strong>in</strong>ed and characterized analogously.<br />
Siep Weiland and Carsten Scherer (DISC) <strong>L<strong>in</strong>ear</strong> <strong>Matrix</strong> <strong>Inequalities</strong> <strong>in</strong> <strong>Control</strong> Class 1 41 / 59
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