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== Definitions ==
In the following definitions, <math>\ \mathbf{x}^\top\ </math> is the transpose of <math>\ \mathbf{x}\ ,</math> <math>\ \mathbf{
=== 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}^\top M\ \mathbf{x} > 0\ </math> for all non-zero <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 '''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
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=== Definitions for complex matrices ===
The following definitions all involve the term <math>\ \mathbf{
An <math>\ n \times n\ </math> Hermitian complex matrix <math>\ M\ </math> is said to be '''positive-definite''' if <math>\ \mathbf{
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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>\
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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{
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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{
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Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.
For complex matrices, the most common definition says that <math>\ M\ </math> is positive-definite if and only if <math>\ \mathbf{z}^* M\ \mathbf{z}\ </math> is real and positive for every non-zero complex column vectors <math>\mathbf{z} ~.</math> This condition implies that <math>M</math> is Hermitian (i.e. its transpose is equal to its conjugate), since <math>\mathbf{z}^* M\ \mathbf{z}</math> being real, it equals its conjugate transpose <math>\ \mathbf{z}^*\ M^*\ \mathbf{z}\ </math> for every <math>\ \mathbf{z}\ ,</math> which implies <math>\ M = M^* ~.</math>
By this definition, a positive-definite ''real'' matrix <math>\ M\ </math> is Hermitian, hence symmetric; and <math>\ \mathbf{z}^\top M\ \mathbf{z}\ </math> is positive for all non-zero ''real'' column vectors <math>\ \mathbf{z} ~.</math> However the last condition alone is not sufficient for <math>\ M\ </math> to be positive-definite. For example, if
<math display="block">\ M = \begin{bmatrix} ~1~ & ~1~ \\ -1~ & ~1~ \end{bmatrix},</math>
then for any real vector <math>\ \mathbf{z}\ </math> with entries <math>\ a\ </math> and <math>\ b\ </math> we have <math>\ \mathbf{z}^\top M\ \mathbf{z} = \left(a + b\right)a + \left(-a + b\right) b = a^2 + b^2\ ,</math> which is always positive if <math>\
<math display="block">\mathbf{z}^* M\ \mathbf{z} = \begin{bmatrix} ~1~ & -i~ \end{bmatrix}\ M\ \begin{bmatrix} ~1~ \\ ~i~ \end{bmatrix} = \begin{bmatrix} ~1 + i~ & ~1 - i~ \end{bmatrix}\ \begin{bmatrix} ~1~ \\ ~i~ \end{bmatrix} = 2 + 2i ~.</math>
which is not real. Therefore, <math>\ M\ </math> is not positive-definite.
On the other hand, for a ''symmetric'' real matrix <math>\ M\ ,</math> the condition "<math>\ \mathbf{z}^\top M\ \mathbf{z} > 0\ </math> for all nonzero real vectors <math>\ \mathbf{z}\ </math>" ''does'' imply that <math>\ M\ </math> is positive-definite in the complex sense.
===Notation===
If a Hermitian matrix <math>\ M\ </math> is positive semi-definite, one sometimes writes <math>\ M \succeq 0\ </math> and if <math>\ M\ </math> is positive-definite one writes <math>\ M \succ 0 ~.</math> To denote that <math>\ M\ </math> is negative semi-definite one writes <math>\ M \preceq 0\ </math> and to denote that <math>\ M\ </math> is negative-definite one writes <math>\ M \prec 0 ~.</math>
The notion comes from [[functional analysis]] where positive semidefinite matrices define [[positive operator]]s. If two matrices <math>\ A\ </math> and <math>\ B\ </math> satisfy <math>\ B - A \succeq 0\ ,</math> we can define a [[Partially ordered set#Non-strict partial order|non-strict partial order]] <math>\ B \succeq A\ </math> that is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]]; It is not a [[total order]], however, as <math>\ B - A\ ,</math> in general, may be indefinite.
A common alternative notation is <math>\ M \geq 0\ ,</math> <math>\ M > 0\ ,</math> <math>\ M \leq 0\ ,</math> and <math>\ M < 0\ </math> for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes [[nonnegative matrix|nonnegative matrices]] (respectively, nonpositive matrices) are also denoted in this way.
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