Semi-Definiteness

123. Semi-Definiteness

Let $𝐱 \in ℝ^{𝑛}$ be any nonzero vector. Look at the quadratic form:

𝐱^{𝑇} 𝐻 𝐱

$𝑞 (𝑥) = 𝐱^{𝑇} 𝐻 𝐱$ $𝐻 𝐱 = 𝜆 𝐱$
	Definition	Eigen Values	Geometry
Positive Definite	$𝑞 (𝐱) > 0 \forall 𝐱 \neq 𝟎$	$\begin{matrix} 𝜆_{𝑖} > 0 \\ 𝐻 ≻ 0 \end{matrix}$	Min (bowl-shaped)
Negative Definite	$𝑞 (𝐱) < 0 \forall 𝐱 \neq 𝟎$	$\begin{matrix} 𝜆_{𝑖} < 0 \\ 𝐻 ≺ 0 \end{matrix}$	Max (inverted bowl)
Positive Semidefinite	$𝑞 (𝐱) \geq 0 \forall 𝐱$	$\begin{matrix} 𝜆_{𝑖} \geq 0 \\ 𝐻 ⪰ 0 \end{matrix}$	Min but flat in some directions
Negative Semidefinite	$𝑞 (𝐱) \leq 0 \forall 𝐱$	$\begin{matrix} 𝜆_{𝑖} \leq 0 \\ 𝐻 ⪯ 0 \end{matrix}$	Max but flat in some directions
Indefinite	some $𝑞 (𝐱_{𝟏}) > 0$ some $𝑞 (𝐱_{𝟐}) < 0$	some $𝜆_{𝑖} > 0$ some $𝜆_{𝑗} < 0$	Saddle Up in some directions Down in others