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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 \(\displaystyle x\)
