Properties of Definite Matrices
Properties
In connection with Definite matrices, the following hold.
(a) For the matrix $A \in M_{n \times n}(\mathbb{C})$, $A^{\ast}A$ is positive semidefinite.
(b) Positive definite matrices and negative definite matrices are always invertible.
Remark
In (a), when $A$ is a real matrix, we may set $A^{\mathsf{T}}A$.
Proof
(a)
For $\mathbf{x} \in \mathbb{C}^{n}$, the following holds.
$$ \mathbf{x}^{\ast} A^{\ast} A \mathbf{x} = (\mathbf{x} A)^{\ast} A \mathbf{x} = (A \mathbf{x}) \cdot (A \mathbf{x}) $$
By the properties of the inner product, the above quantity is greater than or equal to $0$.
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