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Linear Algebra: Difference between revisions
→Definiteness
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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> | |||