Linear Algebra: Difference between revisions
(Created page with "Linear Algebra ==Definiteness== A 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, <mat...") |
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==Definiteness== | ==Definiteness== | ||
A matrix <math>A</math is positive definite if for all vectors <math>x</math>, <math>x^T A x > 0 <math>.<br> | A 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 | If the inequality is weak (<math> \geq </math>) then the matrix is positive semi-definite.<br> | ||
Examples of PSD matrices: | Examples of PSD matrices: | ||
* <math>x x^T<math> is PSD for any vector <math>x</math> | * The identity matrix is PSD | ||
* <math>x x^T</math> is PSD for any vector <math>x</math> | |||
Revision as of 17:16, 31 October 2019
Linear Algebra
Definiteness
A 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.
Examples of PSD matrices:
- The identity matrix is PSD
- \(\displaystyle x x^T\) is PSD for any vector \(\displaystyle x\)