Positive Definite, Semidefinite

Positive Definite, Semidefinite

Definition 1. Let $T \in \mathcal{L}(V)$ where $V$ is a finite-dimensional inner product space, and let $A \in M_{n \times n}(F)$. Then $T$ is called positive definite [positive semidefinite] if $T$ is hermitian and $\langle T(x), x \rangle > 0$ $[\langle T(x), x \rangle \geq 0], \forall x \neq \mathbf{0}$, and $A$ is called positive definite [positive semidefinite] if $L_A$ is positive definite [positive semidefinite].

Theorem 1

Theorem 1. Let $T \in \mathcal{L}(V)$ be hermitian, and let $A = [T]_{\beta}$ where $V$ is $n$-dimensional inner product space and $\beta$ is an orthonormal basis for $V$. Then the followings hold:
(a) $T$ is positive definite [semidefinite] $\iff$ all of its eigenvalues are positive [nonnegative].
(b) $T$ is positive definite [semidefinite] $\iff$ $A$ is positive definite [semidefinite].

Proof. (a) Let $T(x) = \lambda x$ for some $x \neq \mathbf{0}$. Then $\langle T(x), x \rangle = \langle \lambda x, x \rangle$ $= \lambda ||x||^2 > 0 [\geq 0]$. Thus $\lambda > 0 [\geq 0]$.
(b) $(\Longrightarrow)$
$\langle T(x), x \rangle = \langle [T(x)]_{\beta}, [x]_{\beta} \rangle = \langle A[x]_{\beta}, [x]_{\beta} \rangle > [\geq 0]$. Let $y = [x]_{\beta}$. Thus $\langle Ay, y \rangle > 0 [\geq 0]$.
$(\Longleftarrow)$
$\langle Ax, x \rangle = \langle [T]_{\beta}[x]_{\beta}, [x]_{\beta} \rangle = \langle [T(x)]_{\beta}, [x]_{\beta} \rangle = \langle T(x), x \rangle > 0 [\geq 0]$. $\blacksquare$

Theorem 2

Theorem 2. Let $T \in \mathcal{L}(V, W)$ where $V$ and $W$ are finite-dimensional inner product space. Then $T^*T$ and $TT^*$ are positive semidefinite.

Proof. For any $x \neq \mathbf{0}$, $\langle T^*T(x), x \rangle = \langle T(x), T(x) \rangle = ||T(x)||^2 \geq 0$ and $\langle TT^*(x), x \rangle = \langle T^*(x), T^*(x) \rangle = ||T^*(x)||^2 \geq 0$. $\blacksquare$