Matrix Characteristics

Definition of a Positive Definite Matrix

• A symmetric matrix $Q$ is positive definite $⟺Q>0\forall$ eigenvalues $\lambda >0$
• A symmetric matrix $Q$ is positive definite $⟺Q>0\forall$ leading principle minors $>0$

Definition of a Positive Semi-definite Matrix

• A symmetric matrix $Q$ is positive semi - definite $⟺Q>0\forall$ eigenvalues $\lambda \ge 0$
• A symmetric matrix $Q$ is positive semi - definite $⟺Q>0\forall$ leading principle minors $\ge 0$

Quadratic Form

Is used to map functions in the following manner: ${\mathbb{R}}^n\to \mathbb{R}$. In other words it turns $n$-dimensional matrices and turns them into scalars.