Editing Definite matrix (section)

=== Definitions for real matrices ===
An <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''positive-definite''' if <math>\mathbf{x}^\mathsf{T} M\mathbf{x} > 0</math> for all non-zero <math>\mathbf{x}</math> in <math>\mathbb{R}^n.</math> Formally,

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|equation = <math>M \text{ positive-definite} \quad \iff \quad \mathbf{x}^\mathsf{T} M\mathbf{x} > 0 \text{ for all } \mathbf{x} \in \R^n \setminus \{\mathbf{0}\}</math>
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An <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''positive-semidefinite''' or '''non-negative-definite''' if <math>\mathbf{x}^\mathsf{T} M\mathbf{x} \geq 0</math> for all <math>\mathbf{x}</math> in <math>\mathbb{R}^n .</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ positive semi-definite} \quad \iff \quad \mathbf{x}^\mathsf{T} M\mathbf{x} \geq 0 \text{ for all } \mathbf{x} \in \mathbb{R}^n</math>
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An <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''negative-definite''' if <math>\mathbf{x}^\mathsf{T} M\mathbf{x} < 0</math> for all non-zero <math>\mathbf{x}</math> in <math>\R^n.</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ negative-definite} \quad \iff \quad \mathbf{x}^\mathsf{T} M\mathbf{x} < 0 \text{ for all } \mathbf{x} \in \mathbb{R}^n \setminus \{\mathbf{0}\}</math>
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An <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''negative-semidefinite''' or '''non-positive-definite''' if <math>\mathbf{x}^\mathsf{T} M\mathbf{x} \leq 0</math> for all <math>\mathbf{x}</math> in <math>\mathbb{R}^n .</math> Formally,

{{Equation box 1
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|equation = <math>M \text{ negative semi-definite} \quad \iff \quad \mathbf{x}^\mathsf{T} M\mathbf{x} \leq 0 \text{ for all } \mathbf{x} \in \R^n</math>
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An <math>n \times n</math> symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called '''indefinite'''.