123. Semi-Definiteness
123.1. Positive/Negative Definite/Indefinite/Semidefinite Matrices
Let be any nonzero vector. Look at the quadratic form:
|
|
|||
| Definition | Eigen Values | Geometry | |
| Positive Definite |
|
|
Min (bowl-shaped) |
| Negative Definite |
|
|
Max (inverted bowl) |
| Positive Semidefinite |
|
|
Min but flat in some directions |
| Negative Semidefinite |
|
|
Max but flat in some directions |
| Indefinite | some some |
some some |
Saddle Up in some directions Down in others |