Editing Definite matrix (section)

=== Definitions for complex matrices ===
The following definitions all involve the term <math>\mathbf{z}^* M\mathbf{z}.</math> Notice that this is always a real number for any Hermitian square matrix <math>M.</math>

An <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''positive-definite''' if <math>\mathbf{z}^* M\mathbf{z} > 0</math> for all non-zero <math>\mathbf{z}</math> in <math>\mathbb{C}^n .</math> Formally,

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|equation = <math>M \text{ positive-definite} \quad \iff \quad \mathbf{z}^* M\mathbf{z} > 0 \text{ for all } \mathbf{z} \in \mathbb{C}^n \setminus \{ \mathbf{0} \}</math>
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An <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''positive semi-definite''' or '''non-negative-definite''' if <math>\mathbf{z}^* M\mathbf{z} \geq 0</math> for all <math>\mathbf{z}</math> in <math>\mathbb{C}^n .</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ positive semi-definite} \quad \iff \quad \mathbf{z}^* M\mathbf{z} \geq 0 \text{ for all } \mathbf{z} \in \mathbb{C}^n</math>
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An <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''negative-definite''' if <math>\mathbf{z}^* M\mathbf{z} < 0</math> for all non-zero <math>\mathbf{z}</math> in <math>\mathbb{C}^n .</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ negative-definite} \quad \iff \quad \mathbf{z}^* M\mathbf{z} < 0 \text{ for all } \mathbf{z} \in \mathbb{C}^n \setminus \{\mathbf{0}\}</math>
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An <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''negative semi-definite''' or '''non-positive-definite''' if <math>\mathbf{z}^* M\mathbf{z} \leq 0</math> for all <math>\mathbf{z}</math> in <math>\mathbb{C}^n .</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ negative semi-definite} \quad \iff \quad \mathbf{z}^* M\mathbf{z} \leq 0 \text{ for all } \mathbf{z} \in \mathbb{C}^n</math>
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An <math>n \times n</math> Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called '''indefinite'''.