View source for Definiteness of a matrix

{{hatnote|Not to be confused with [[Positive matrix]] and [[Totally positive matrix]].}}

In [[linear algebra]], a [[symmetric matrix|symmetric]] <math>n \times n</math> [[real number|real]] [[matrix (mathematics)|matrix]] <math>M</math> is said to be '''positive definite''' if the scalar <math>z^\textsf{T}Mz</math> is strictly positive for every non-zero column [[vector (mathematics)|vector]] <math>z</math> of <math>n</math> real numbers. Here <math>z^\textsf{T}</math> denotes the [[transpose]] of <math>z</math>.<ref>{{cite journal|doi=10.1002/9780470173862.app3 | title=Appendix C: Positive Semidefinite and Positive Definite Matrices | journal=Parameter Estimation for Scientists and Engineers | pages=259–263}}</ref>  When interpreting <math>Mz</math> as the output of an operator, <math>M</math>, that is acting on an input, <math>z</math>, the property of positive definiteness implies that the output always has a positive [[inner product]] with the input, as often observed in physical processes.

More generally, a complex <math>n\times n</math> [[Hermitian matrix]] <math>M</math> is said to be '''positive definite''' if the scalar <math>z^* Mz</math> is strictly positive for every non-zero column vector <math>z</math> of <math>n</math> complex numbers. Here <math>z^*</math> denotes the [[conjugate transpose]] of <math>z</math>. Note that <math>z^* Mz</math> is automatically real since <math>M</math> is Hermitian.

'''Positive semi-definite''' matrices are defined similarly, except that the above scalars  <math>z^\textsf{T}Mz</math> or <math>z^* Mz</math> must be positive ''or zero'' (i.e. non-negative). '''Negative definite''' and '''negative semi-definite''' matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is called '''indefinite'''.

The matrix <math>M</math> is positive definite if and only if the [[bilinear form]] <math>\langle z, w\rangle = z^\textsf{T} Mw</math> is [[definite bilinear form|positive definite]] (and similarly for a positive definite [[sesquilinear form]] in the complex case). This is a coordinate realization of an [[inner product]] on a [[vector space]].<ref>{{cite journal|last=Stewart|first=J.|date=1976|title=Positive definite functions and generalizations, an historical survey|journal=Rocky Mountain J. Math|volume=6|issue=3|pages=409–434|doi=10.1216/RMJ-1976-6-3-409}}</ref>

Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.

== Definitions ==
In the following definitions, <math>x^\textsf{T}</math> is the transpose of <math>x</math>, <math>x^*</math> is the [[conjugate transpose]] of <math>x</math> and <math>\mathbf{0}</math> denotes the ''n''-dimensional zero-vector.

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

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A <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''positive semidefinite''' or '''non-negative definite''' if <math>x^\textsf{T} Mx \geq 0</math> for all <math>x</math> in <math>\mathbb{R}^n</math>. Formally,

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A <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''negative definite''' if <math>x^\textsf{T} Mx < 0</math> for all non-zero <math>x</math> in <math>\mathbb{R}^n</math>. Formally,

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A <math>n \times n</math> symmetric real matrix <math>M</math> is said to be '''negative semidefinite''' or '''non-positive definite''' if <math>x^\textsf{T} Mx \leq 0</math> for all <math>x</math> in <math>\mathbb{R}^n</math>. Formally,

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A <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>x^* Mx</math>. Notice that this is always a real number for any Hermitian square matrix <math>M</math>.

A <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''positive definite''' if <math>x^* Mx > 0</math> for all non-zero <math>x</math> in <math>\mathbb{C}^n</math>. Formally,

{{Equation box 1
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A <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''positive semi-definite''' or '''non-negative definite''' if <math>x^* Mx \geq 0</math> for all <math>x</math> in <math>\mathbb{C}^n</math>. Formally,

{{Equation box 1
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A <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''negative definite''' if <math>x^* Mx < 0</math> for all non-zero <math>x</math> in <math>\mathbb{C}^n</math>. Formally,

{{Equation box 1
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A <math>n \times n</math> Hermitian complex matrix <math>M</math> is said to be '''negative semi-definite''' or '''non-positive definite''' if <math>x^* Mx \leq 0</math> for all <math>x</math> in <math>\mathbb{C}^n</math>. Formally,

{{Equation box 1
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|equation = <math>M \text{ negative semi-definite} \quad \iff \quad x^* Mx \leq 0 \text{ for all } x \in \mathbb{C}^n</math>
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A <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>z^* Mz</math> is real and positive for all non-zero ''complex'' column vectors <math>z</math>".  This condition implies that <math>M</math> is Hermitian (i.e. its transpose is equal to its conjugate). To see this, consider the matrices <math>A = \tfrac{1}{2} \left(M + M^*\right)</math> and <math>B = \tfrac{1}{2i} \left(M - M^*\right)</math>, so that <math>M = A + iB</math> and  <math>z^* Mz = z^* Az + iz^* Bz</math>. The matrices <math>A</math> and <math>B</math> are Hermitian, therefore <math>z^* Az</math> and <math>z^* Bz</math> are individually real. If <math>z^* Mz</math> is real, then <math>z^* Bz</math> must be zero for all <math>z</math>. Then <math>B</math> is the zero matrix and <math>M = A</math>, proving that <math>M</math> is Hermitian.

By this definition, a positive definite ''real'' matrix <math>M</math> is Hermitian, hence symmetric; and <math>z^\textsf{T} Mz</math> is positive for all non-zero ''real'' column vectors <math>z</math>.  However the last condition alone is not sufficient for <math>M</math> to be positive definite. For example, if
: <math>M = \begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix},</math>

then for any real vector <math>z</math> with entries <math>a</math> and <math>b</math> we have <math>z^\textsf{T} Mz = (a - b)a + (a + b)b = a^2 + b^2</math>, which is always positive if <math>z</math> is not zero. However, if <math>z</math> is the complex vector with entries <math>1</math> and <math>i</math>, one gets

:<math>z^* M z = [1, -i] M [1, i]^\textsf{T} = [1 + i, 1 - i] [1, i]^\textsf{T} = 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>z^\textsf{T} Mz > 0</math> for all nonzero real vectors <math>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.

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.

== Examples ==
{{unordered list
| The [[identity matrix]] <math>I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math> is positive definite (and as such also positive semi-definite). It is a real symmetric matrix, and, for any non-zero column vector ''z'' with real entries ''a'' and ''b'', one has
: <math> z^\textsf{T}Iz = \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^*Iz = \begin{bmatrix} \overline{a} & \overline{b} \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} a \\ b\end{bmatrix} = \overline{a}a + \overline{b}b = |a|^2 + |b|^2</math>.

Either way, the result is positive since <math>z</math> is not the zero vector (that is, at least one of <math>a</math> and <math>b</math> is not zero).
| The real symmetric matrix
: <math>M = \begin{bmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}</math>

is positive definite since for any non-zero column vector ''z'' with entries ''a'', ''b'' and ''c'', we have
: <math>\begin{align}
  z^\textsf{T} Mz = \left(z^\textsf{T}M\right) z
    &= \begin{bmatrix} (2a - b) & (-a + 2b - c) & (-b + 2c) \end{bmatrix}
         \begin{bmatrix} a \\ b \\ c \end{bmatrix} \\
    &= (2a - b)a + (-a + 2b - c)b + (-b + 2c)c \\
    &= 2a^2 - ba - ab + 2b^2 - cb - bc + 2c^2 \\
    &= 2a^2 - 2ab + 2b^2 - 2bc + 2c^2 \\
    &=  a^2 + a^2 - 2ab + b^2 + b^2- 2bc + c^2 + c^2 \\
    &=  a^2 + (a - b)^2 + (b - c)^2 + c^2
\end{align}</math>

This result is a sum of squares, and therefore non-negative; and is zero only if <math>a = b = c = 0</math>, that is, when ''z'' is the zero vector.
| For any real [[Invertible matrix|invertible]] matrix <math>A</math>, the product <math>A^\textsf{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^\textsf{T} A^\textsf{T} Az = (Az)^\textsf{T}(Az) = \|Az\|^2 > 0,</math> since the invertibility of matrix <math>A</math> means that <math>Az \neq 0.</math>
| The example <math>M</math> above shows that a matrix in which some elements are negative may still be positive definite. Conversely, a matrix whose entries are all positive is not necessarily positive definite, as for example
: <math>N = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix},</math>

for which <math>\begin{bmatrix} -1 & 1 \end{bmatrix}N\begin{bmatrix} -1 & 1 \end{bmatrix}^\textsf{T} = -2 < 0.</math>
}}

== Eigenvalues ==
Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix]].

* <math>M</math> is positive definite if and only if all of its eigenvalues are positive.
* <math>M</math> is positive semi-definite if and only if all of its eigenvalues are non-negative.
* <math>M</math> is negative definite if and only if all of its eigenvalues are negative
* <math>M</math> is negative semi-definite if and only if all of its eigenvalues are non-positive.
* <math>M</math> is indefinite if and only if it has both positive and negative eigenvalues.

Let <math>P^{-1} DP</math> be an [[eigendecomposition of a matrix|eigendecomposition]] of <math>M</math>, where <math>P</math> is a [[unitary matrix|unitary complex matrix]] whose rows comprise an [[orthonormal basis]] of [[eigenvector]]s of <math>M</math>, and <math>D</math> is a ''real'' [[diagonal matrix]] whose [[main diagonal]] contains the corresponding [[eigenvalue]]s. The matrix <math>M</math> may be regarded as a diagonal matrix <math>D</math> that has been re-expressed in coordinates of the basis <math>P</math>. In particular, the one-to-one change of variable <math>y = Pz</math> shows that <math>z^* Mz</math> is real and positive for any complex vector <math>z</math> if and only if <math>y^* Dy</math> is real and positive for any <math>y</math>; in other words, if <math>D</math> is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of  <math>M</math>—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 <math>M</math> is available.

== Connections ==
A general purely [[quadratic form|quadratic real function]] <math>f(\mathbf{x})</math> on <math>n</math> real variables <math>x_1, \ldots, x_n</math> can always be written as <math>\mathbf{x}^\textsf{T} M \mathbf{x}</math> where <math>\mathbf{x}</math> is the column vector with those variables, and <math>M</math> is a symmetric real matrix. Therefore, the matrix being positive definite means that <math>f</math> has a unique minimum (zero) when <math>\mathbf{x}</math> is zero, and is strictly positive for any other <math>\mathbf{x}</math>.

More generally, a twice-differentiable real function <math>f</math> on <math>n</math> real variables has local minimum at arguments <math>x_1, \ldots, x_n</math> if its [[gradient]] is zero and its [[Hessian matrix|Hessian]] (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices.

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 function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.

== Characterizations ==
Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix]]. The following properties are equivalent to <math>M</math> being positive definite:
; The associated sesquilinear form is an inner product: The [[sesquilinear form]] defined by <math>M</math> is the function <math>\langle \cdot, \cdot\rangle</math> from <math>\mathbb{C}^n \times \mathbb{C}^n</math> to <math>\mathbb{C}^n</math> such that <math>\langle x, y \rangle := y^*M x</math> for all <math>x</math> and <math>y</math> in <math>\mathbb{C}^n</math>, where <math>y^*</math> is the conjugate transpose of <math>y</math>. For any complex matrix <math>M</math>, this form is linear in <math>x</math> and semilinear in <math>y</math>.  Therefore, the form is an [[inner product]] on <math>\mathbb{C}^n</math> if and only if <math>\langle z, z \rangle</math> is real and positive for all nonzero <math>z</math>; that is if and only if <math>M</math> is positive definite. (In fact, every inner product on <math>\mathbb{C}^n</math> arises in this fashion from a Hermitian positive definite matrix.)
; It is the Gram matrix of a set of linearly independent vectors: Let <math>x_1, \ldots, x_n</math> be a list of <math>n</math> [[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]] <math>M</math> of those vectors, defined by <math>M_{ij} = \langle x_i, x_j \rangle</math>, is always positive definite.  Conversely, if <math>M</math> is positive definite, it has an eigendecomposition <math>P^{-1} DP</math> where <math>P</math> is unitary, <math>D</math> diagonal, and all diagonal elements <math>D_{ii} = \lambda_i</math> of <math>D</math> are real and positive. Let <math>E</math> 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 EP = (EP)^* EP</math>. Now we let <math>x_1, \ldots, x_n</math> be the columns of <math>EP</math>.  These vectors are linearly independent, and by the above <math>M</math> is their Gram matrix, under the standard inner product of <math>\mathbb{C}^n</math>, namely <math>\langle x_i, x_j \rangle = x_i^\ast x_j</math>.
; Its leading principal minors are all positive: The ''k''th [[minor (linear algebra)|leading principal minor]] of a matrix <math>M</math> is the [[determinant]] of its upper-left <math>k \times k</math> 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 <math>k</math>, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive.  This condition can be checked each time a new row <math>k</math> of the triangular matrix is obtained.

== Quadratic forms, convexity, optimization ==
The (purely) [[quadratic form]] associated with a real <math>n \times n</math> matrix <math>M</math> is the function <math>Q : \mathbb{R}^n \mapsto \mathbb{R}</math> such that <math>Q(x) = x^\textsf{T} Mx</math> for all <math>x</math>. <math>M</math> can be assumed symmetric by replacing it with <math>\tfrac{1}{2} \left(M + M^\textsf{T}\right)</math>. 

A symmetric matrix <math>M</math> is positive definite if and only if its quadratic form is a [[strictly convex function]].

More generally, any [[quadratic function]] from <math>\mathbb{R}^n</math> to <math>\mathbb{R}</math> can be written as <math>x^\textsf{T} Mx + x^\textsf{T} b + c</math> where <math>M</math> is a symmetric <math>n \times n</math> matrix, <math>b</math> is a real <math>n</math>-vector, and <math>c</math> a real constant.  This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if <math>M</math> is positive definite<!-- Citation needed  The Bernstein-Toupin reference covers this. -->. For this reason, positive definite matrices play an important role in [[optimization (mathematics)|optimization]] problems.

== Simultaneous diagonalization ==
A symmetric matrix and another symmetric and positive definite matrix can be [[diagonalizable matrix#Simultaneous diagonalization|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.

Let <math>M</math> be a symmetric and <math>N</math> a symmetric and positive definite matrix. Write the generalized eigenvalue equation as <math>(M - \lambda N)x = 0</math> where we impose that <math>x</math> be normalized, i.e. <math>x^\textsf{T} Nx = 1</math>. Now we use [[Cholesky decomposition]] to write the inverse of <math>N</math> as <math>Q^\textsf{T} Q</math>. Multiplying by <math>Q</math> and letting <math>x = Q^\textsf{T} y</math>, we get <math>Q(M - \lambda N)Q^\textsf{T} y = 0</math>, which can be rewritten as <math>\left(QMQ^\textsf{T}\right)y = \lambda y</math> where <math>y^\textsf{T} y = 1</math>. Manipulation now yields <math>MX = NX\Lambda</math> where <math>X</math> is a matrix having as columns the generalized eigenvectors and <math>\Lambda</math> is a diagonal matrix of the generalized eigenvalues.  Now premultiplication with <math>X^\textsf{T}</math> gives the final result: <math>X^\textsf{T} MX = \Lambda</math> and <math>X^\textsf{T} NX = I</math>, but note that this is no longer an orthogonal diagonalization with respect to the inner product where <math>y^\textsf{T} y = 1</math>. In fact, we diagonalized <math>M</math> with respect to the inner product induced by <math>N</math>.

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.<ref>{{harvtxt|Horn|Johnson|1985}}, p. 218 ff.</ref>

== Properties ==
===Induced partial ordering===
For arbitrary square matrices <math>M</math>, <math>N</math> we write <math>M \ge N</math> if <math>M - N \ge 0</math> i.e., <math>M - N</math> 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 <math>M > N</math>.
The ordering is called the [[Loewner order]].

===Inverse of positive definite matrix===
Every positive definite matrix is [[invertible matrix|invertible]] and its inverse is also positive definite.<ref>{{harvtxt|Horn|Johnson|1985}}, p. 397</ref> If <math>M \geq N > 0</math> then <math>N^{-1} \geq M^{-1} > 0</math>.<ref>{{harvtxt|Horn|Johnson|1985}}, Corollary 7.7.4(a)</ref> Moreover, by the [[min-max theorem]], the ''k''th largest eigenvalue of <math>M</math> is greater than the ''k''th largest eigenvalue of <math>N</math>.

=== Scaling ===
If <math>M</math> is positive definite and <math>r > 0</math> is a real number, then <math>rM</math> is positive definite.<ref name="HJobs713">{{harvtxt|Horn|Johnson|1985}}, Observation 7.1.3</ref>

=== Addition ===
If <math>M</math> and <math>N</math> are positive definite, then the sum <math>M + N</math> is also positive definite.<ref name="HJobs713"/>

===Multiplication===
* If <math>M</math> and <math>N</math> are positive definite, then the products <math>MNM</math> and <math>NMN</math> are also positive definite. If <math>MN = NM</math>, then <math>MN</math> is also positive definite.

* If <math>M</math> is positive semidefinite, then <math>Q^\textsf{T} MQ</math> is positive semidefinite. If <math>M</math> is positive definite and <math>Q</math> has full rank, then <math>Q^\textsf{T} MQ</math> is positive definite.<ref>
{{Cite book
 | last1=Horn | first1=Roger A.
 | last2=Johnson | first2=Charles R.
 | title=Matrix Analysis
 | edition=2nd
 | chapter=7.1 Definitions and Properties| publisher=[[Cambridge University Press]]
 | isbn=978-0-521-83940-2
 | year=2013
 | page=431
}} "'''Observation 7.1.8''' Let <math>A \in M_n</math> be Hermitian and let <math>C \in M_{n,m}</math>:
*Suppose that {{mvar|A}} is positive semidefinite. Then <math>C^* AC</math> is positive semidefinite, <math>\operatorname{nullspace}C^* AC) = \operatorname{nullspace}(AC)</math>, and <math>\operatorname{rank}(C^* AC) = \operatorname{rank}(AC)^*</math>
*Suppose that {{mvar|A}} is positive definite. Then <math>\operatorname{rank}(C^* AC) = \operatorname{rank}(C)</math>, and <math>C^* AC</math> is positive definite [[if and only if]] rank({{mvar|C}}) {{=}} {{mvar|m}}"</ref>

===Cholesky decomposition===
For any matrix <math>A</math>, the matrix <math>A^* A</math> is positive semidefinite, and <math>\operatorname{rank}(A) = \operatorname{rank}(A^* A)</math>. Conversely, any Hermitian positive semi-definite matrix <math>M</math> can be written as <math>M = LL^*</math>, where <math>L</math> is lower triangular; this is the [[Cholesky decomposition]]. If <math>M</math> is not positive definite, then some of the diagonal elements of <math>L</math> may be zero.

A hermitian matrix <math>M</math> is positive definite if and only if it has a unique Cholesky decomposition, i.e. the matrix <math>M</math> is positive definite if and only if there exists a unique lower triangular matrix <math>L</math>, with real and strictly positive diagonal elements, such that <math>M = LL^*</math>.

===Square root===
A matrix <math>M</math> is positive semi-definite if and only if there is a positive semi-definite matrix <math>B</math> with <math>B^2 = M</math>. This matrix <math>B</math> is unique,<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 7.2.6 with <math>k = 2</math></ref> is called the [[square root of a matrix|square root]] of <math>M</math>, and is denoted with <math>B = M^\frac{1}{2}</math> (the square root <math>B</math> is not to be confused with the matrix <math>L</math> in the [[Cholesky factorization]] <math>M = LL^*</math>, which is also sometimes called the square root of <math>M</math>).

If <math> M > N > 0 </math> then <math>M^\frac{1}{2} > N^\frac{1}{2} > 0</math>.

===Submatrices===
Every principal submatrix of a positive definite matrix is positive definite.

===Trace===
The diagonal entries <math>m_{ii}</math> of a positive definite matrix are real and non-negative. As a consequence the [[trace (linear algebra)|trace]], <math>\operatorname{tr}(M) \ge 0</math>. Furthermore,<ref>{{harvtxt|Horn|Johnson|1985}}, p. 398</ref> since every principal sub-matrix (in particular, 2-by-2) is positive definite,
: <math>\left|m_{ij}\right| \leq \sqrt{m_{ii}m_{jj}} \leq \frac{m_{ii} + m_{jj}}{2} \quad \forall i, j</math>

and thus
: <math> \max_{i,j} \left|m_{ij}\right| \leq \max_i\left|m_{ii}\right|</math>

===Hadamard product===
If <math>M, N \geq 0</math>, although <math>MN</math> is not necessary positive semidefinite, the [[Hadamard product (matrices)|Hadamard product]] <math>M \circ N \geq 0</math> (this result is often called the [[Schur product theorem]]).<ref>{{harvtxt|Horn|Johnson|1985}}, Theorem 7.5.3</ref>

Regarding the Hadamard product of two positive semidefinite matrices <math>M = (m_{ij}) \geq 0</math>, <math>N \geq 0</math>, there are two notable inequalities:
* 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>
* <math>\det(M \circ N) \geq \det(M) \det(N)</math>.<ref name=styan1973>{{harvtxt|Styan|1973}}{{fcn|date=September 2019}}</ref>

===Kronecker product===
If <math>M, N \geq 0</math>, although <math>MN</math> is not necessary positive semidefinite, the [[Kronecker product]] <math>M \otimes N \geq 0</math>.

===Frobenius product===
If <math>M, N \geq 0</math>, although <math>MN</math> is not necessary positive semidefinite, the [[Matrix multiplication#Frobenius product|Frobenius product]] <math>M : N \geq 0</math> (Lancaster–Tismenetsky, ''The Theory of Matrices'', p.&nbsp;218).

===Convexity===
The set of positive semidefinite symmetric matrices is [[convex set|convex]].  That is, if <math>M</math> and <math>N</math> are positive semidefinite, then for any <math>\alpha</math> between 0 and 1, <math>\alpha M + (1 - \alpha) N</math> is also positive semidefinite.  For any vector <math>x</math>:
:<math>x^\textsf{T} \left(\alpha M + (1 - \alpha)N\right)x = \alpha x^\textsf{T} Mx + (1 - \alpha) x^\textsf{T} Nx \geq 0.</math>

This property guarantees that [[semidefinite programming]] problems converge to a globally optimal solution.

=== Relation with cosine ===
The positive-definiteness of a matrix <math>A
</math> expresses that the angle <math>\theta</math> between any vector <math>x</math> and its image <math>Ax</math> is always <math>-\pi / 2 < \theta < +\pi / 2 </math>:

<math>\cos(\theta)=\frac{x^{T}Ax}{\left\Vert x \right\Vert \left\Vert Ax \right\Vert}=\frac{<x,Ax>}{\left\Vert x \right\Vert \left\Vert Ax \right\Vert} ,  \theta=\theta(x,Ax)=\widehat{x,Ax}= the \ angle \ between \ x \ and \ Ax</math>

===Further properties===
{{ordered list
| If <math>M</math> is a symmetric [[Toeplitz matrix]], i.e. the entries <math>m_{ij}</math> are given as a function of their absolute index differences: <math>m_{ij} = h(|i-j|)</math>, and the ''strict'' inequality
<math>\sum\nolimits_{j \neq 0} \left|h(j)\right| < h(0)</math>
holds, then <math>M</math> is ''strictly'' positive definite.
| Let <math>M > 0</math> and <math>N</math> Hermitian. If <math>MN + NM \ge 0</math> (resp., <math>MN + NM > 0</math>) then <math>N \ge 0</math> (resp., <math>N > 0</math>).<ref>
 {{Cite book
  | title=Positive Definite Matrices
  | last=Bhatia | first=Rajendra
  | publisher=Princeton University Press
  | year=2007
  | isbn=978-0-691-12918-1
  | location=Princeton, New Jersey
  | pages=8
 }}</ref>
| If <math>M > 0</math> is real, then there is a <math>\delta > 0</math> such that <math>M>\delta I</math>, where <math>I</math> is the [[identity matrix]].
| If <math>M_k</math> denotes the leading <math>k \times k</math> minor, <math>\det\left(M_k\right)/\det\left(M_{k-1}\right)</math> is the ''k''th pivot during [[LU decomposition]].
| A matrix is negative definite if its ''k-''th order leading [[principal minor]] is negative when <math>k</math> is odd, and positive when <math>k</math> is even.
| A matrix <math>M</math> 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.
}}

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.

=== Block matrices ===
A positive <math>2n \times 2n</math> matrix may also be defined by [[block matrix|blocks]]:
: <math> M =  \begin{bmatrix} A & B \\ C & D \end{bmatrix}</math>

where each block is <math>n \times n</math>. By applying the positivity condition, it immediately follows that <math>A</math> and <math>D</math> are hermitian, and <math>C = B^*</math>.

We have that <math>z^* Mz \ge 0</math> for all complex <math>z</math>, and in particular for <math>z = [v, 0]^\textsf{T}</math>. Then
: <math>\begin{bmatrix} v^* & 0 \end{bmatrix} \begin{bmatrix} A & B \\ B^* & D \end{bmatrix} \begin{bmatrix} v \\ 0 \end{bmatrix} = v^* Av \ge 0.</math>

A similar argument can be applied to <math>D</math>, and thus we conclude that both <math>A</math> and <math>D</math> must be positive definite matrices, as well.

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]].

== Extension for non-Hermitian square matrices ==
The definition of positive definite can be generalized by designating any complex matrix <math>M</math> (e.g. real non-symmetric) as positive definite if <math>\Re\left(z^* Mz\right) > 0</math> for all non-zero complex vectors <math>z</math>, where <math>\Re(c)</math> denotes the real part of a complex number <math>c</math>.<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> Only the Hermitian part <math>\tfrac{1}{2}\left(M + M^*\right)</math> determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, If <math>x</math> and <math>M</math> are real, we have <math>x^\textsf{T} M x > 0</math> for all real nonzero vectors <math>x</math> if and only if the symmetric part <math>\tfrac{1}{2}\left(M + M^\textsf{T}\right)</math> is positive definite in the narrower sense. It is immediately clear that <math>x^\textsf{T} M x=x_iM_{ij}x_j</math>is insensitive to transposition of M.

''Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite.'' For example, the matrix <math>M = \left[\begin{smallmatrix} 4 & 9 \\ 1 & 4 \end{smallmatrix}\right]</math> has positive eigenvalues yet is not positive definite; in particular a negative value of <math>x^\textsf{T} Mx</math> is obtained with the choice <math>x = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right] </math> (which is the eigenvector associated with the negative eigenvalue of the symmetric part of <math>M</math>).

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.

== Applications ==
=== Heat conductivity matrix ===
Fourier's law of heat conduction, giving heat flux <math>q</math> in terms of the temperature gradient <math>g = \nabla T</math> is written for anisotropic media as <math>q = -Kg</math>, in which <math>K</math> is the symmetric [[thermal conductivity]] matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient <math>g</math> always points from cold to hot, the heat flux <math>q</math> is expected to have a negative inner product with <math>g</math> so that <math>q^\textsf{T}g < 0</math>. Substituting Fourier's law then gives this expectation as <math>g^\textsf{T}Kg > 0</math>, implying that the conductivity matrix should be positive definite.

== See also ==
*[[Cholesky decomposition]]
*[[Covariance matrix]]
*[[M-matrix]]
*[[Positive-definite function]]
*[[Positive-definite kernel]]
*[[Schur complement]]
*[[Square root of a matrix]]
*[[Sylvester's criterion]]
*[[Symmetric matrix]]
*[[Numerical range]]

== Notes ==
{{Reflist|40em}}

== References ==
* {{Cite book | 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}}
* {{cite book |authorlink=Rajendra Bhatia |first=Rajendra |last=Bhatia |title=Positive definite matrices |series=Princeton Series in Applied Mathematics |year=2007 |isbn=978-0-691-12918-1 }}
* {{cite journal |last=Bernstein |first=B. |last2=Toupin |first2=R. A. |year=1962 |title=Some Properties of the Hessian Matrix of a Strictly Convex Function |journal=[[Journal für die reine und angewandte Mathematik]] |volume=210 |issue= |pages=67–72 |doi=10.1515/crll.1962.210.65 }}

== External links ==
* {{springer|title=Positive-definite form|id=p/p073880}}
* [http://mathworld.wolfram.com/PositiveDefiniteMatrix.html Wolfram MathWorld: Positive Definite Matrix]

{{Matrix classes}}

{{DEFAULTSORT:Positive-Definite Matrix}}
[[Category:Matrices]]

[[de:Definitheit#Definitheit von Matrizen]]