definition of positive-definite Hermitian matrix
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of Hermitian matrix.
Target Context
- The reader will have a definition of positive-definite Hermitian matrix.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*M\): \(\in \{\text{ the } n \times n \text{ Hermitian matrices }\}\)
//
Conditions:
\(\forall v \in \{\text{ the } n \times 1 \text{ complex matrices }\}\)
(
\(0 \le v^t M v\)
\(\land\)
\(v = 0 \iff v^t M v = 0\)
)
//
2: Note
For example, \(I\) is a positive-definite Hermitian matrix, which is a proof that there is at least 1 positive-definite Hermitian matrix.
More generally, any positive diagonal matrix, which means that each diagonal element is positive, is a positive-definite Hermitian matrix.
