1143: Positive-Definite Hermitian Matrix

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definition of positive-definite Hermitian matrix

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Topics

About: matrices space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of Hermitian matrix.

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Target Context

• The reader will have a definition of positive-definite Hermitian matrix.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$\ast M$: $\in \{\text{ the }n\times n\text{ Hermitian matrices }\}$
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Conditions:
$\mathrm{\forall }v\in \{\text{ the }n\times 1\text{ complex matrices }\}$
(
$0\le v^{t}Mv$
$\wedge$
$v=0⟺v^{t}Mv=0$
)
//

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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.

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References

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