Linear Algebra: Difference between revisions

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A square matrix <math>A</math> is positive definite if for all vectors <math>x</math>, <math>x^T A x > 0 </math>.<br>
A square matrix <math>A</math> is positive definite if for all vectors <math>x</math>, <math>x^T A x > 0 </math>.<br>
If the inequality is weak (<math> \geq </math>) then the matrix is positive semi-definite.<br>
If the inequality is weak (<math> \geq </math>) then the matrix is positive semi-definite.<br>
===Properties===
* If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.<br>
* If <math>A</math> is PSD then <math>A^T</math> is PSD.
* The sum of PSD matrices is PSD.
===Examples===
Examples of PSD matrices:
Examples of PSD matrices:
* The identity matrix is PSD
* The identity matrix is PSD
* <math>x x^T</math> is PSD for any vector <math>x</math>
* <math>x x^T</math> is PSD for any vector <math>x</math>
If the determinant of every upper-left submatrix is positive then the matrix is positive-definite.<br>
If <math>A</math> is PSD then <math>A^T</math> is PSD.
The sum of PSD matrices is PSD.

Latest revision as of 17:18, 31 October 2019

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 \(\displaystyle x\)