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 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 | 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: | ||
* <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> | |||
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 \(\displaystyle x\)