Positive Definiteness
A matrix is positive definite if all eigenvalues are positive. A matrix is positive semi-definite if all eigenvalues are positive or zero.
Contents
Description
A positive definite matrix, by definition, is a symmetric matrix whose eigenvalues are all positive.
Such a matrix has several useful properties:
Always invertible
- All pivots are positive
The determinant is positive, and all subdeterminants are also positive
'Semi-definite' is a slight modification, allowing 0 as an eigenvalue.
Operations
If A is positive definite, then so is cA for any real scalar c.
If A and B are both positive definite...
so is A + B.
so is ABA.
so is BAB.
If A and B are both positive semi-definite, then so is A + B.
If A is definite and B is semi-definite, then A + B is definite.