MATH319 Slides 121 Conclusion of proof 123 Lyapunov’s criterion

122 Positive definite matrices

We write $z=(z_{j})_{j=1}^{n}$ and $w=(w_{j})_{j=1}^{n}$ , and introduce the inner product $\langle z,w\rangle=\sum_{j=1}^{n}z_{j}\bar{w}_{j}$ . We define the adjoint of a $n\times n$ matrix $A=[a_{jk}]$ by $A^{\dagger}=[\bar{a}_{kj}]$ , interchanging the rows and columns and taking the complex conjugate. If $A$ is real then $A^{\dagger}=A^{T}$ , the transpose. An $n\times n$ complex matrix $K$ is said to be positive definite if $K=K^{\dagger}$ and $\langle Kz,z\rangle>0$ for all $z\in{\bf C}^{n}$ such that $z\neq 0$ . Beware that the product of positive definite matrices is generally not positive definite.

Lemma (MATH220 Theorem 5.15)

Let $K$ be a $(n\times n)$ complex matrix such that $K=K^{\dagger}$ . Then the following are equivalent:

(i) $\langle Kz,z\rangle>0$ for all $z\in{\bf C}^{n}$ such that $z\neq 0$ ;

(ii) the eigenvalues $\kappa_{j}$ of $K$ are all real and $\kappa_{j}>0$ for all $j$ ;

(iii) the leading principal minors $\Delta_{j}$ of $K$ are all positive, so $\Delta_{j}>0$ for all $j$ .