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Kabelsalat22 (talk | contribs) →Definitions: Fixing instances of Equation box 1 template appearance in dark mode |
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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}^\top M\ \mathbf{x} \geq 0\ </math> for all <math>\ \mathbf{x}\ </math> in <math>\ \mathbb{R}^n ~.</math> Formally,
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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}^\top M\ \mathbf{x} < 0\ </math> for all non-zero <math>\ \mathbf{x}\ </math> in <math>\ \R^n ~.</math> Formally,
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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}^\top M\ \mathbf{x} \leq 0\ </math> for all <math>\ \mathbf{x}\ </math> in <math>\ \mathbb{R}^n ~.</math> Formally,
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An <math>n \times n</math> symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called '''indefinite'''.
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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,
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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,
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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,
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An <math>\ n \times n\ </math> Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called '''indefinite'''.
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