Editing Definite matrix (section)

== Definitions ==
In the following definitions, <math>\mathbf{x}^\mathsf{T}</math> is the transpose of <math>\mathbf{x},</math> <math>\mathbf{z}^*</math> is the [[conjugate transpose]] of <math>\mathbf{z},</math> and <math>\mathbf{0}</math> denotes the {{nobr|{{mvar|n}} dimensional}} zero-vector.

=== 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,

{{Equation box 1
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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'''.

=== 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,

{{Equation box 1
|indent =
|title=
|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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|border colour = #0073CF
|background colour=var(--background-color-success-subtle,#d5fdf4)}}

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
|indent =
|title=
|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>
|cellpadding= 6
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|border colour = #0073CF
|background colour=var(--background-color-success-subtle,#d5fdf4)}}

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
|indent =
|title=
|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>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=var(--background-color-success-subtle,#d5fdf4)}}

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
|indent =
|title=
|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>
|cellpadding= 6
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|border colour = #0073CF
|background colour=var(--background-color-success-subtle,#d5fdf4)}}

An <math>n \times n</math> Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called '''indefinite'''.

=== Consistency between real and complex definitions ===
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}^\mathsf{T} 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}^\mathsf{T} M\mathbf{z} = \left(a + b\right)a + \left(-a + b\right) b = a^2 + b^2,</math> which is always positive if <math>\mathbf{z}</math> is not zero. However, if <math>\mathbf{z}</math> is the complex vector with entries {{math|1}} and {{tmath| i }}, one gets

<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}^\mathsf{T} 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.