Positive Definite Matrix

Definition

With $n\times n$ matrix $M$ that has Quadratic Form $Q_M(h)=h^{T}Mh$ $M$ is positive definite if:

• $Q_M(h)>0,\forall h\ne 0$ and $Q_M(h)=0$ equivalent to $h=0$ $M$ is negative definite if:
• $Q_M(h)<0,\forall h\ne 0$ and $Q_M(h)=0$ equivalent to $h=0$

Alternate Characterization

• All Principal Minors of the matrix are positive $⟹$ positive definite matrix
• All Principal Minors of the matrix are negative $⟹$ negative definite matrix
• Principal Minors are mixed in Polarity $⟹$ indefinite matrix