Linear Algebra

Linear Algebra

Definiteness

A square matrix $A$ is positive definite if for all vectors $x$, $x^{T}Ax>0$.
If the inequality is weak ($\geq$) then the matrix is positive semi-definite.

Properties

• If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.
  
• If $A$ is PSD then $A^{T}$ is PSD.
• The sum of PSD matrices is PSD.

Examples

Examples of PSD matrices:

• The identity matrix is PSD
• $xx^{T}$ is PSD for any vector $x$