Linear Algebra


Linear Algebra

Definiteness

A square matrix $$\displaystyle A$$ is positive definite if for all vectors $$\displaystyle x$$, $$\displaystyle x^T A x \gt 0$$.
If the inequality is weak ($$\displaystyle \geq$$) 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 $$\displaystyle A$$ is PSD then $$\displaystyle A^T$$ is PSD.
• The sum of PSD matrices is PSD.

Examples

Examples of PSD matrices:

• The identity matrix is PSD
• $$\displaystyle x x^T$$ is PSD for any vector $$\displaystyle x$$