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| {{hatnote|Not to be confused with [[Positive matrix]] and [[Totally positive matrix]].}}
| | This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users. |
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| In [[linear algebra]], a [[symmetric matrix|symmetric]] {{nowrap|''n'' × ''n''}} [[real number|real]] [[matrix (mathematics)|matrix]] ''M'' is said to be '''positive definite''' if ''z''<sup>T</sup>''Mz'' is positive for every non-zero column [[vector (mathematics)|vector]] ''z'' of ''n'' real numbers. Here ''z''<sup>T</sup> denotes the [[transpose]] of ''z''.
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| More generally, an {{nowrap|''n'' × ''n''}} [[Hermitian matrix]] ''M'' is said to be '''positive definite''' if ''z*Mz'' is real and positive for all non-zero column vectors ''z'' of ''n'' complex numbers. Here ''z*'' denotes the [[conjugate transpose]] of ''z''.
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| The '''negative definite''', '''positive semi-definite''', and '''negative semi-definite''' matrices are defined in the same way, except that the expression ''z''<sup>T</sup>''Mz'' or ''z*Mz'' is required to be always negative, non-negative, and non-positive, respectively.
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| | :<math forcemathmode="mathml">E=mc^2</math> |
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| Positive definite matrices are closely related to [[definite bilinear form|positive-definite]] symmetric [[bilinear form]]s (or [[sesquilinear form]]s in the complex case), and to [[inner product]]s of vector spaces.<ref>[http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.rmjm/1250130219&page=record Stewart, J. (1976). Positive definite functions and generalizations, an historical survey. Rocky Mountain J. Math, 6(3).]</ref>
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| Some authors use more general definitions of "positive definite" that include some non-symmetric real matrices, or non-Hermitian complex ones.
| | '''source''' |
| | :<math forcemathmode="source">E=mc^2</math> --> |
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| == Examples == | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
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| *The [[identity matrix]] <math> I = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} </math> is positive definite. Seen as a real matrix, it is symmetric, and, for any non-zero column vector ''z'' with real entries ''a'' and ''b'', one has <math> z^{\mathrm{T}}I z = \begin{bmatrix} a & b\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix} a \\ b\end{bmatrix}= a^2 + b^2</math>. Seen as a complex matrix, for any non-zero column vector ''z'' with complex entries ''a'' and ''b'' one has <math> z^*I z = \begin{bmatrix} a^* & b^*\end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix} a \\ b\end{bmatrix}=a^*a +b^*b = |a|^2 + |b|^2</math>. Either way, the result is positive since ''z'' is not the zero vector (that is, at least one of ''a'' and ''b'' is not zero).
| | ==Demos== |
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| * The real symmetric matrix
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| ::<math> M = \begin{bmatrix} 2&-1&0\\-1&2&-1\\0&-1&2 \end{bmatrix} </math>
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| :is positive definite since for any non-zero column vector ''z'' with entries ''a'', ''b'' and ''c'', we have
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| ::<math>\begin{align} z^{\mathrm{T}}M z = (z^{\mathrm{T}}M) z &= \begin{bmatrix} (2a-b)&(-a+2b-c)&(-b+2c) \end{bmatrix} \begin{bmatrix} a\\b\\c \end{bmatrix} \\
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| &= 2{a}^2 - 2ab + 2{b}^2 - 2bc + 2{c}^2 \\
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| &= {a}^2+(a - b)^{2} + (b - c)^{2}+{c}^2
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| \end{align} </math>
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| :This result is a sum of squares, and therefore non-negative; and is zero only if ''a'' = ''b'' = ''c'' = 0, that is, when ''z'' is zero.
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| *The real symmetric matrix | | * accessibility: |
| | ** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]] |
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| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| ::<math>N = \begin{bmatrix} 1 & 2 \\ 2 & 1\end{bmatrix} </math>
| | ==Test pages == |
| :is not positive definite. If ''z'' is the vector <math>\begin{bmatrix} 1\\ -1\end{bmatrix}</math>, one has <math> z^{\mathrm{T}}N z = \begin{bmatrix} 1 & -1\end{bmatrix} \begin{bmatrix} 1 & 2 \\ 2 & 1\end{bmatrix} \begin{bmatrix} 1 \\ -1\end{bmatrix}=\begin{bmatrix} -1 & 1\end{bmatrix} \begin{bmatrix} 1 \\ -1\end{bmatrix}=-2 \not > 0.</math>
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| *For any real non-singular matrix <math>A</math>, the product <math> A^T A </math> is a positive definite matrix. A simple proof is that for any non-zero vector <math> z </math>, the condition <math> z^T A^T A z = \| A z \|_2^2 > 0, </math> since the non-singularity of matrix <math>A</math> means that <math>Az \neq 0.</math>
| | To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages: |
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| The examples ''M'' and ''N'' above show that a matrix in which some elements are negative may still be positive-definite, and conversely a matrix whose entries are all positive may not be positive definite.
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
| | | *[[Url2Image|Url2Image (private Wikis only)]] |
| ==Connections==
| | ==Bug reporting== |
| The general purely [[quadratic form|quadratic real function]] ''f''(''z'') on ''n'' real variables ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'' can always be written as ''z''<sup>T</sup>''Mz'' where ''z'' is the column vector with those variables, and ''M'' is a symmetric real matrix. Therefore, the matrix being positive definite means that ''f'' has a unique minimum (zero) when ''z'' is zero, and is strictly positive for any other ''z''.
| | If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de . |
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| More generally, a twice-differentiable real function ''f'' on ''n'' real variables has an isolated local minimum at arguments ''z''<sub>1</sub>, ..., ''z<sub>n</sub>'' if its [[gradient]] is zero and its [[Hessian matrix|Hessian]] (the matrix of all second derivatives) is positive definite at that point. Similar statements can be made for negative definite and semi-definite matrices.
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| In [[statistics]], the [[covariance matrix]] of a [[multivariate probability distribution]] is always positive semi-definite; and it is positive definite unless one variable is an exact linear combination of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.
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| == Characterizations ==
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| Let ''M'' be an ''n'' × ''n'' [[Hermitian matrix]]. The following properties are equivalent to ''M'' being positive definite:
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| #'''All its [[eigenvalues]] are positive.''' Let ''P''<sup>−1</sup>''DP'' be an [[eigendecomposition of a matrix|eigendecomposition]] of ''M'', where ''P'' is a [[unitary matrix|unitary complex matrix]] whose rows comprise an [[orthonormal basis]] of [[eigenvector]]s of ''M'', and ''D'' is a ''real'' [[diagonal matrix]] whose [[main diagonal]] contains the corresponding [[eigenvalue]]s. The matrix ''M'' may be regarded as a diagonal matrix ''D'' that has been re-expressed in coordinates of the basis ''P''. In particular, the one-to-one change of variable ''y'' = ''Pz'' shows that ''z*Mz'' is real and positive for any complex vector ''z'' if and only if ''y*Dy'' is real and positive for any ''y''; in other words, if ''D'' is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of ''M''—is positive. Since the [[spectral theorem]] guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using [[Descartes' rule of signs|Descartes' rule of alternating signs]] when the [[characteristic polynomial]] of a real, symmetric matrix ''M'' is available.
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| #'''The associated sesquilinear form is an inner product.''' The [[sesquilinear form]] defined by ''M'' is the function <math>\langle \cdot,\cdot\rangle</math> from '''C'''<sup>''n''</sup> × '''C'''<sup>''n''</sup> to '''C''' such that <math>\langle x,y\rangle := y^*M x</math> for all ''x'' and ''y'' in '''C'''<sup>''n''</sup>, where ''y<sup>*</sup>'' is the complex conjugate of ''y''. For any complex matrix ''M'', this form is linear in each argument separately. Therefore the form is an [[inner product]] on '''C'''<sup>''n''</sup> if and only if <math>\langle z,z\rangle</math> is real and positive for all nonzero ''z''; that is if and only if ''M'' is positive definite. (In fact, every inner product on '''C'''<sup>''n''</sup> arises in this fashion from a Hermitian positive definite matrix.)
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| # '''It is the Gram matrix of linearly independent vectors.''' Let <math>x_1,\ldots,x_n</math> be a list of ''n'' [[linearly independent]] vectors of some [[complex vector space]] with an inner product <math>\langle \cdot,\cdot\rangle</math>. It can be verified that the [[Gram matrix]] ''M'' of those vectors, defined by <math>M_{ij} = \langle x_i, x_j\rangle</math>, is always positive definite. Conversely, if ''M'' is positive definite, it has an eigendecomposition ''P''<sup>−1</sup>''DP'' where ''P'' is unitary, ''D'' diagonal, and all diagonal elements D<sub>''ii''</sub> = λ<sub>''i''</sub> of ''D'' are real and positive. Let ''E'' be the real diagonal matrix with entries <math> E_{ii} = \sqrt{\lambda_i} </math> so <math> E^2 = D </math>; then <math> P^{-1}DP = P^*DP = P^*E E P = (EP)^* EP. </math> Now we let <math>x_1,\ldots,x_n</math> be the columns of ''EP''. These vectors are linearly independent, and by the above ''M'' is their Gram matrix, under the standard inner product of '''C'''<sup>''n''</sup>, namely <math>\langle x_i, x_j\rangle = x_i^* x_j</math>
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| # '''Its leading principal minors are all positive.''' The ''k''th [[minor (linear algebra)|leading principal minor]] of a matrix ''M'' is the [[determinant]] of its upper-left ''k'' by ''k'' sub-matrix. It turns out that a matrix is positive definite if and only if all these determinants are positive. This condition is known as [[Sylvester's criterion]], and provides an efficient test of positive-definiteness of a symmetric real matrix. Namely, the matrix is reduced to an [[upper triangular matrix]] by using [[elementary row operations]], as in the first part of the [[Gaussian elimination]] method, taking care to preserve the sign of its determinant during [[pivot element|pivoting]] process. Since the ''k''th leading principal minor of a triangular matrix is the product of its diagonal elements up to row ''k'', Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. This condition can be checked each time a new row ''k'' of the triangular matrix is obtained.
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| #'''It has a unique Cholesky decomposition.''' The matrix ''M'' is positive definite if and only if there exists a unique lower triangular matrix ''L'', with real and strictly positive diagonal elements, such that ''M'' = ''LL*''. This factorization is called the [[Cholesky decomposition]] of ''M''.
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| ==Quadratic forms==
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| The (purely) [[quadratic form]] associated with a real matrix ''M'' is the function ''Q'' : '''R'''<sup>''n''</sup> → '''R''' such that ''Q''(''x'') = ''x''<sup>T</sup>''Mx'' for all ''x''. It turns out that the matrix ''M'' is positive definite if and only if it is symmetric and its quadratic form is a [[strictly convex function]].
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| More generally, any [[quadratic function]] from '''R'''<sup>''n''</sup> to '''R''' can be written as ''x''<sup>T</sup>''Mx'' + ''x''<sup>T</sup>''b'' + ''c'' where ''M'' is a symmetric ''n'' × ''n'' matrix, ''b'' is a real ''n''-vector, and ''c'' a real constant. This quadratic function is strictly convex when ''M'' is positive definite, and hence has a unique finite global minimum, if and only if ''M'' is positive definite. For this reason, positive definite matrices play an important role in [[optimization (mathematics)|optimization]] problems.
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| == Simultaneous diagonalization ==
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| A symmetric, and a symmetric and positive-definite matrix can be simultaneously diagonalized, although not necessarily via a [[Matrix similarity|similarity transformation]]. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.
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| Let ''M'' be a symmetric and ''N'' a symmetric and positive-definite matrix. Write the generalized eigenvalue equation as (''M''−λ''N'')''x'' = 0 where we impose that ''x'' be normalized, i.e. ''x''<sup>T</sup>''Nx'' = 1. Now we use [[Cholesky decomposition]] to write the inverse of ''N'' as ''Q''<sup>T</sup>''Q''. Multiplying by ''Q'' and ''Q''<sup>T</sup>, we get ''Q''(''M''−λ''N'')''Q''<sup>T</sup>''x'' = 0, which can be rewritten as (''QMQ''<sup>T</sup>)''y'' = λ''y'' where ''y''<sup>T</sup>''y'' = 1. Manipulation now yields ''MX'' = ''NX''Λ where ''X'' is a matrix having as columns the generalized eigenvectors and Λ is a diagonal matrix with the generalized eigenvalues. Now premultiplication with ''X''<sup>T</sup> gives the final result: ''X''<sup>T</sup>''MX'' = Λ and ''X''<sup>T</sup>''NX'' = ''I'', but note that this is no longer an orthogonal diagonalization.
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| Note that this result does not contradict what is said on simultaneous diagonalization in the article [[Diagonalizable matrix#Simultaneous diagonalization|Diagonalizable matrix]], which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other. For this result see Horn&Johnson, 1985, page 218 and following.
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| == Negative-definite, semidefinite and indefinite matrices ==<!-- [[Positive-semidefinite matrix]] and [[Positive semi-definite matrix]] redirect here -->
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| A [[Hermitian matrix]] is negative-definite, negative-semidefinite, or positive-semidefinite if and only if all of its [[eigenvalues]] are negative, non-positive, or non-negative, respectively.
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| === Negative-definite ===
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| The {{nowrap|''n'' × ''n''}} Hermitian matrix ''M'' is said to be ''negative-definite'' if
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| :<math>x^{*} M x < 0\,</math><!-- The \, is to keep the formula rendered as PNG instead of HTML. Please don't remove it.-->
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| for all non-zero ''x'' in '''C'''<sup>''n''</sup> (or, all non-zero ''x'' in '''R'''<sup>''n''</sup> for the real matrix), where ''x*'' is the [[conjugate transpose]] of ''x''.
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| A matrix is negative definite if its ''k''th order leading principal minor is negative when ''k'' is odd, and positive when ''k'' is even.
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| === Positive-semidefinite === | |
| M is called ''positive-semidefinite'' (or sometimes ''nonnegative-definite'') if
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| :<math>x^{*} M x \geq 0</math>
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| for all ''x'' in '''C'''<sup>''n''</sup> (or, all ''x'' in '''R'''<sup>''n''</sup> for the real matrix).
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| A matrix ''M'' is positive-semidefinite if and only if it arises as the [[Gram matrix]] of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent.
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| For any matrix ''A'', the matrix ''A*A'' is positive semidefinite, and rank(''A'') = rank(''A*A'').
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| Conversely, any Hermitian positive semi-definite matrix ''M'' can be written as ''M'' = ''LL*'', where ''L'' is lower triangular; this is the [[Cholesky decomposition]]. If ''M'' is not positive definite, then some of the diagonal elements of ''L'' may be zero.
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| A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and -1.
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| === Negative-semidefinite ===
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| It is called ''negative-semidefinite'' if
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| :<math>x^{*} M x \leq 0</math> | |
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| for all ''x'' in '''C'''<sup>''n''</sup> (or, all ''x'' in '''R'''<sup>''n''</sup> for the real matrix).
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| === Indefinite ===
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| A Hermitian matrix which is neither positive definite, negative definite, positive-semidefinite, nor negative-semidefinite is called ''indefinite''. Indefinite matrices are also characterized by having both positive and negative eigenvalues.
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| == Further properties ==
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| If ''M'' is a Hermitian positive-semidefinite matrix, one sometimes writes ''M'' ≥ 0 and if ''M'' is positive-definite one writes ''M'' > 0.<ref>This may be confusing, as sometimes [[nonnegative matrix|nonnegative matrices]] are also denoted in this way. A common alternative notation is <math>M \succeq 0</math> and <math> M \succ 0</math> for positive semidefinite and positive definite matrices, respectively.</ref> The notion comes from [[functional analysis]] where positive-semidefinite matrices define [[positive operator]]s.
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| For arbitrary square matrices ''M'', ''N'' we write ''M'' ≥ ''N'' if ''M'' − ''N'' ≥ 0; i.e., ''M'' − ''N'' is positive semi-definite. This defines a [[partially ordered set|partial ordering]] on the set of all square matrices. One can similarly define a strict partial ordering ''M'' > ''N''.
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| # Every positive definite matrix is [[invertible matrix|invertible]] and its inverse is also positive definite.<ref>{{harvtxt|Horn|Johnson|1985}}, p. 397</ref> If ''M'' ≥ ''N'' > 0 then ''N''<sup>−1</sup> ≥ ''M''<sup>−1</sup> > 0, and {{sqrt|''M''}} > {{sqrt|''N''}} > 0.<ref>{{harvtxt|Horn|Johnson|1985}}, Corollary 7.7.4(a)</ref> Moreover, by the [[min-max theorem]], the ''k''th largest eigenvalue of ''M'' is greater than the ''k''th largest eigenvalue of ''N''
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| # If ''M'' is positive definite and ''r'' > 0 is a real number, then ''rM'' is positive definite.<ref name="HJobs713">{{harvtxt|Horn|Johnson|1985}}, Observation 7.1.3</ref> If ''M'' and ''N'' are positive definite, then the sum ''M'' + ''N''<ref name="HJobs713"/> and the products ''MNM'' and ''NMN'' are also positive definite. If ''MN'' = ''NM'', then ''MN'' is also positive definite.
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| # Every principal submatrix of a positive definite matrix is positive definite.
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| # ''Q<sup>T</sup> M Q'' is non-negative definite. If ''Q'' is invertible, then ''Q<sup>T</sup> M Q'' is positive definite. Note that ''Q<sup>-1</sup> M Q'' need not to be positive definite.
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| # The determinant of ''M'' is bounded by the product of its diagonal elements.
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| # The diagonal entries ''m<sub>ii</sub>'' are real and non-negative. As a consequence the [[trace (linear algebra)|trace]], tr(''M'') ≥ 0. Furthermore,<ref>{{harvtxt|Horn|Johnson|1985}}, p. 398</ref> since every principal sub matrix (in particular, 2-by-2) is positive definite,
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| #::<math>|m_{ij}| \leq \sqrt{m_{ii}m_{jj}} \leq \frac{m_{ii}+m_{jj}}{2}</math>
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| #:and thus
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| #::<math> \max |m_{ij}| \leq \max|m_{ii}|</math>
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| # A matrix ''M'' is positive semi-definite if and only if there is a positive semi-definite matrix ''B'' with ''B''<sup>2</sup> = ''M''. This matrix ''B'' is unique,<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 7.2.6 with ''k'' = 2</ref> is called the [[square root of a matrix|square root]] of ''M'', and is denoted with ''B'' = ''M''<sup>1/2</sup> (the square root ''B'' is not to be confused with the matrix ''L'' in the [[Cholesky factorization]] ''M'' = ''LL*'', which is also sometimes called the square root of ''M''). If ''M'' > ''N'' > 0 then ''M''<sup>1/2</sup> > ''N''<sup>1/2</sup> > 0.
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| # If ''M'' is a symmetric matrix of the form ''m<sub>ij</sub>'' = ''m''(''i''−''j''), and the ''strict'' inequality holds
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| #::<math>\sum\nolimits_{j\neq 0}|m(j)|< m(0)</math>
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| #:then ''M'' is ''strictly'' positive definite.
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| # Let ''M'' > 0 and ''N'' Hermitian. If ''MN'' + ''NM'' ≥ 0 (resp., ''MN'' + ''NM'' > 0) then ''N'' ≥ 0 (resp., ''N'' > 0).{{citation needed|date=November 2010}}
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| # If ''M'' > 0 is real, then there is a δ > 0 such that ''M'' > δI, where ''I'' is the [[identity matrix]].
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| # If ''M<sub>k</sub>'' denotes the leading ''k'' by ''k'' minor, <math> \det(M_k)/\det(M_{k-1}) </math> is the ''k''th pivot during [[LU decomposition]].
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| # The set of positive semidefinite symmetric matrices is [[convex set|convex]]. That is, if ''M'' and ''N'' are positive semidefinite, then for any α between 0 and 1, α''M'' + (1−α)''N'' is also positive semidefinite. For any vector ''x'':
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| #:: <math> x^\top (\alpha M + (1-\alpha) N ) x = \alpha x^\top M x + (1-\alpha) x^\top N x \geq 0. </math>
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| #:This property guarantees that [[semidefinite programming]] problems converge to a globally optimal solution.
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| # If ''M,N'' ≥ 0, although ''MN'' is not necessary positive-semidefinite, the [[Kronecker product]] ''M'' ⊗ ''N'' ≥ 0, the [[Hadamard product (matrices)|Hadamard product]] ''M'' ○ ''N'' ≥ 0 (this result is often called the [[Schur product theorem]]).,<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 7.5.3</ref> and the [[Matrix multiplication#Frobenius product|Frobenius product]] ''M'' : ''N'' ≥ 0 (Lancaster-Tismenetsky, The Theory of Matrices, p. 218).
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| # Regarding the Hadamard product of two positive-semidefinite matrices ''M'' = (''m<sub>ij</sub>'') ≥ 0, ''N'' ≥ 0, there are two notable inequalities:
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| #*Oppenheim's inequality: <math>\det(M \circ N) \geq \det (N) \prod\nolimits_{i} m_{ii}.</math><ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 7.8.6</ref>
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| #*det(''M'' ○ ''N'') ≥ det(''M'') det(''N'').<ref name=styan1973>{{Harvard citations | last=Styan | year=1973}}</ref>
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| == Block matrices ==
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| A positive 2''n'' × 2''n'' matrix may also be defined by [[block matrix|blocks]]:
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| :<math> M = \begin{bmatrix} A & B \\ C & D \end{bmatrix} </math>
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| where each block is ''n'' × ''n''. By applying the positivity condition, it immediately follows that ''A'' and ''D'' are hermitian, and ''C'' = ''B*''.
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| We have that ''z*Mz'' ≥ 0 for all complex ''z'', and in particular for ''z'' = ( ''v'', 0)<sup>T</sup>. Then
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| :<math> \begin{bmatrix} v^* & 0 \end{bmatrix} \begin{bmatrix} A & B \\ B^* & D \end{bmatrix} \begin{bmatrix} v \\ 0 \end{bmatrix} = v^* A v \ge 0. </math>
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| A similar argument can be applied to ''D'', and thus we conclude that both ''A'' and ''D'' must be positive definite matrices, as well.
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| Converse results can be proved with stronger conditions on the blocks, for instance using the [[Schur complement#Schur complement condition for positive definiteness|Schur complement]].
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| == On the definition ==
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| ===Consistency between real and complex definitions===
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| Since every real matrix is also a complex matrix, the definitions of "positive definite" for the two classes must agree.
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| For complex matrices, the most common definition says that "''M'' is positive definite if and only if ''z*Mz'' is real and positive for all non-zero ''complex'' column vectors ''z''". This condition implies that ''M'' is Hermitian, that is, its transpose is equal to its conjugate. To see this, consider the matrices ''A'' = (''M''+''M*'')/2 and ''B'' = (''M''−''M*'')/(2''i''), so that ''M'' = ''A''+''iB'' and ''z*Mz'' = ''z*Az'' + ''iz*Bz''. The matrices ''A'' and ''B'' are Hermitian, therefore ''z*Az'' and ''z*Bz'' are individually real. If ''z*Mz'' is real, then ''z*Bz'' must be zero for all ''z''. Then ''B'' is the zero matrix and ''M'' = ''A'', proving that ''M'' is Hermitian.
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| By this definition, a positive definite ''real'' matrix ''M'' is Hermitian, hence symmetric; and ''z''<sup>T</sup>''Mz'' is positive for all non-zero ''real'' column vectors ''z''". However the last condition alone is not sufficient for ''M'' to be positive definite. For example, if
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| :<math>M = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix},</math>
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| then for any real vector ''z'' with entries ''a'' and ''b'' we have ''z''<sup>T</sup>''Mz'' = (''a''−''b'')''a'' + (''a''+''b'')''b'' = ''a''<sup>2</sup> + ''b''<sup>2</sup>, which is always positive if ''z'' is not zero. However, if ''z'' is the complex vector with entries 1 and ''i'', one gets
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| :''z*Mz'' = [1, −''i'']''M''[1, ''i'']<sup>T</sup> = [1+''i'', 1−''i''][1, ''i'']<sup>T</sup> = 2 + 2''i'',
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| which is not real. Therefore, ''M'' is not positive definite.
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| On the other hand, for a ''symmetric'' real matrix ''M'', the condition "''z''<sup>T</sup>''Mz'' > 0 for all nonzero real vectors ''z''" ''does'' imply that ''M'' is positive definite in the complex sense.
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| ===Extension for non symmetric matrices===
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| Some authors choose to say that a complex matrix ''M'' is positive definite if Re(''z*Mz'') > 0 for all non-zero complex vectors ''z'', where Re(''c'') denotes the real part of a complex number ''c''.<ref name="mathw">Weisstein, Eric W. ''[http://mathworld.wolfram.com/PositiveDefiniteMatrix.html Positive Definite Matrix.]'' From ''MathWorld--A Wolfram Web Resource''. Accessed on 2012-07-26</ref> This weaker definition encompasses some non-Hermitian complex matrices, including some non-symmetric real ones, such as <math> \bigl[ \begin{smallmatrix} 1 & 1 \\ -1 & 1 \end{smallmatrix} \bigr] </math>.
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| Indeed, with this definition, a real matrix is positive definite if and only if ''z''<sup>T</sup>''Mz'' > 0 for all nonzero real vectors ''z'', even if ''M'' is not symmetric.
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| In general, we have Re(''z*Mz'') > 0 for all complex nonzero vectors ''z'' if and only if the Hermitian part (''M'' + ''M*'')/2 of ''M'' is positive definite in the narrower sense. Similarly, we have ''x''<sup>T</sup>''Mx'' > 0 for all real nonzero vectors ''x'' if and only if the symmetric part (''M'' + ''M''<sup>T</sup>)/2 of ''M'' is positive definite in the narrower sense.
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| In summary, the distinguishing feature between the real and complex case is that, a [[Bounded operator|bounded]] positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the [[polarization identity]]. That is no longer true in the real case.
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| == See also ==
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| *[[Cholesky decomposition]]
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| *[[Covariance matrix]]
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| *[[M-matrix]]
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| *[[Positive-definite function]]
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| *[[Positive-definite kernel]]
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| *[[Schur complement]]
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| *[[Square root of a matrix]]
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| *[[Sylvester's criterion]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| * {{Citation | last1=Horn | first1=Roger A. | last2=Johnson | first2=Charles R. | title=Matrix Analysis | publisher=[[Cambridge University Press]] | isbn=978-0-521-38632-6 | year=1990}}.
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| * [[Rajendra Bhatia]]. ''Positive definite matrices''. Princeton Series in Applied Mathematics, 2007. ISBN 978-0-691-12918-1.
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| == External links ==
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| * {{springer|title=Positive-definite form|id=p/p073880}}
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| * [http://mathworld.wolfram.com/PositiveDefiniteMatrix.html Wolfram MathWorld: Positive Definite Matrix]
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| {{DEFAULTSORT:Positive-Definite Matrix}}
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| [[Category:Matrices]]
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| [[de:Definitheit#Definitheit von Matrizen]]
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