Linear Algebra

From David's Wiki
Revision as of 13:18, 31 October 2019 by David (talk | contribs) (→‎Definiteness)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
\( \newcommand{\P}[]{\unicode{xB6}} \newcommand{\AA}[]{\unicode{x212B}} \newcommand{\empty}[]{\emptyset} \newcommand{\O}[]{\emptyset} \newcommand{\Alpha}[]{Α} \newcommand{\Beta}[]{Β} \newcommand{\Epsilon}[]{Ε} \newcommand{\Iota}[]{Ι} \newcommand{\Kappa}[]{Κ} \newcommand{\Rho}[]{Ρ} \newcommand{\Tau}[]{Τ} \newcommand{\Zeta}[]{Ζ} \newcommand{\Mu}[]{\unicode{x039C}} \newcommand{\Chi}[]{Χ} \newcommand{\Eta}[]{\unicode{x0397}} \newcommand{\Nu}[]{\unicode{x039D}} \newcommand{\Omicron}[]{\unicode{x039F}} \DeclareMathOperator{\sgn}{sgn} \def\oiint{\mathop{\vcenter{\mathchoice{\huge\unicode{x222F}\,}{\unicode{x222F}}{\unicode{x222F}}{\unicode{x222F}}}\,}\nolimits} \def\oiiint{\mathop{\vcenter{\mathchoice{\huge\unicode{x2230}\,}{\unicode{x2230}}{\unicode{x2230}}{\unicode{x2230}}}\,}\nolimits} \)

Linear Algebra


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.


  • 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 of PSD matrices:

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