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{{hatnote|Not to be confused with [[Positive matrix]] and [[Totally positive matrix]].}}
{{use dmy dates|date=June 2024}}
In [[mathematics]], a symmetric matrix <math>M</math> with [[real number|real]] entries is '''positive-definite''' if the real number <math>\mathbf{x}^\top M \mathbf{x}</math> is positive for every nonzero real [[column vector]] <math>\mathbf{x},</math> where <math>\mathbf{x}^\top</math> is the [[row vector]] [[transpose]] of <math>\mathbf{x}.</math><ref>
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}} Print ed. {{ISBN|9780470147818}}
</ref>
More generally, a [[Hermitian matrix]] (that is, a [[complex matrix]] equal to its [[conjugate transpose]]) is '''positive-definite''' if the real number <math>\mathbf{z}^* M \mathbf{z}</math> is positive for every nonzero complex column vector <math>\mathbf{z},</math> where <math>\mathbf{z}^*</math> denotes the conjugate transpose of <math>\mathbf{z}
'''Positive semi-definite''' matrices are defined similarly, except that the scalars <math>\mathbf{x}^\top M \mathbf{x}</math> and <math>\mathbf{z}^* M \mathbf{z}</math> are required to be positive ''or zero'' (that is, nonnegative). '''Negative-definite''' and '''negative semi-definite''' matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called ''indefinite''.
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* <math>M</math> is symmetric or Hermitian, and all its [[eigenvalue]]s are real and positive.
* <math>M</math> is symmetric or Hermitian, and all its leading [[principal minor]]s are positive.
* There exists an [[invertible matrix]] <math>B</math> with conjugate transpose <math>B^*</math> such that <math>M = B^* B
A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.
Positive-definite and positive-semidefinite real matrices are at the basis of [[convex optimization]], since, given a [[function of several real variables]] that is twice [[differentiable function|differentiable]], then if its [[Hessian matrix]] (matrix of its second partial derivatives) is positive-definite at a point <math>p,</math> then the function is [[convex function|convex]] near {{mvar|p}}, and, conversely, if the function is convex near <math>p,</math> then the Hessian matrix is positive-semidefinite at <math>p
The set of positive definite matrices is an [[Open set|open]] [[convex cone]], while the set of positive semi-definite matrices is a [[closed set|closed]] convex cone.<ref>
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=== 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
{{Equation box 1
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|background colour=var(--background-color-success-subtle,#d5fdf4)}}
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
{{Equation box 1
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=== 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
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,
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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}
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>\mathbf{z}</math> is not zero. However, if <math>\mathbf{z}</math> is the complex vector with entries {{math|1}} and <math>i,</math> one gets
<math display="block">\mathbf{z}^* M\mathbf{z} = \begin{bmatrix} 1 & -i \end{bmatrix}M\begin{bmatrix}
which is not real. Therefore, <math>M</math> is not positive-definite.
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===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
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.
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<math display="block"> \mathbf{z}^\top I\mathbf{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 display="block">\mathbf{z}^*I\mathbf{z} = \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>\mathbf 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
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* <math>M</math> is indefinite if and only if it has both positive and negative eigenvalues.
Let <math>P D P^{-1}</math> be an [[eigendecomposition of a matrix|eigendecomposition]] of <math>M,</math> where <math>P</math> is a [[unitary matrix|unitary complex matrix]] whose columns 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 (eigenvectors) basis <math>P
With this in mind, the one-to-one change of variable <math>\mathbf{y} = P\mathbf{z}</math> shows that <math>\mathbf{z}^* M\mathbf{z}</math> is real and positive for any complex vector <math>\mathbf{z}</math> if and only if <math>\mathbf{y}^* D \mathbf{y}</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.
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of a matrix <math>B</math> with its [[conjugate transpose]].
When <math>M</math> is real, <math>B</math> can be real as well and the decomposition can be written as <math display="block">M = B^\top B
<math>M</math> is positive definite if and only if such a decomposition exists with <math>B</math> [[Invertible matrix|invertible]].
More generally, <math>M</math> is positive semidefinite with rank <math>k</math> if and only if a decomposition exists with a <math>k \times n</math> matrix <math>B</math> of full row rank (i.e. of rank <math>k</math>).
Moreover, for any decomposition <math>M = B^* B,</math> <math>\operatorname{rank}(M) = \operatorname{rank}(B)
{{math proof | proof =
If <math>M = B^* B,</math> then <math>x^* M x = (x^* B^*) (B x) = \|B x \|^2 \geq 0,</math> so <math>M</math> is positive semidefinite.
If moreover <math>B</math> is invertible then the inequality is strict for <math>x \neq 0,</math> so <math>M</math> is positive definite.
If <math>B</math> is <math>k \times n</math> of rank <math>k,</math> then <math>\operatorname{rank}(M) = \operatorname{rank}(B^*) = k
In the other direction, suppose <math>M</math> is positive semidefinite.
Since <math>M</math> is Hermitian, it has an [[Eigendecomposition of a matrix#Decomposition for special matrices|eigendecomposition]] <math>M = Q^{-1} D Q</math> where <math>Q</math> is [[unitary matrix|unitary]] and <math>D</math> is a diagonal matrix whose entries are the eigenvalues of <math>M</math>
Since <math>M</math> is positive semidefinite, the eigenvalues are non-negative real numbers, so one can define <math>D^{\frac{1}{2}}</math> as the diagonal matrix whose entries are non-negative square roots of eigenvalues.
Then <math>M = Q^{-1} D Q = Q^* D Q = Q^* D^{\frac{1}{2}} D^{\frac{1}{2}} Q = Q^* D^{\frac{1}{2}*} D^{\frac{1}{2}} Q = B^* B</math> for <math>B = D^{\frac{1}{2}} Q
If moreover <math>M</math> is positive definite, then the eigenvalues are (strictly) positive, so <math>D^{\frac{1}{2}}</math> is invertible, and hence <math>B = D^{\frac{1}{2}} Q</math> is invertible as well.
If <math>M</math> has rank <math>k,</math> then it has exactly <math>k</math> positive eigenvalues and the others are zero, hence in <math>B = D^{\frac{1}{2}} Q</math> all but <math>k</math> rows are all zeroed.
Cutting the zero rows gives a <math>k \times n</math> matrix <math>B'</math> such that <math>B'^* B' = B^* B = M
}}
The columns <math>b_1, \dots, b_n</math> of <math>B</math> can be seen as vectors in the [[complex vector space|complex]] or [[real vector space]] <math>\mathbb{R}^k,</math> respectively.
Then the entries of <math>M</math> are [[inner product]]s (that is [[dot product]]s, in the real case) of these vectors
<math display="block">M_{ij} = \langle b_i, b_j\rangle
In other words, a Hermitian matrix <math>M</math> is positive semidefinite if and only if it is the [[Gram matrix]] of some vectors <math>b_1, \dots, b_n
It is positive definite if and only if it is the Gram matrix of some [[linearly independent]] vectors.
In general, the rank of the Gram matrix of vectors <math>b_1, \dots, b_n</math> equals the dimension of the space [[Linear span|spanned]] by these vectors.<ref>{{harvtxt|Horn|Johnson|2013}}, p. 441, Theorem 7.2.10</ref>
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The decomposition is not unique:
if <math>M = B^* B</math> for some <math>k \times n</math> matrix <math>B</math> and if <math>Q</math> is any [[unitary matrix|unitary]] <math>k \times k</math> matrix (meaning <math>Q^* Q = Q Q^* = I</math>),
then <math>M = B^* B = B^* Q^* Q B = A^* A</math> for <math>A = Q B
However, this is the only way in which two decompositions can differ: The decomposition is unique up to [[unitary transformation]]s.
More formally, if <math>A</math> is a <math>k \times n</math> matrix and <math>B</math> is a <math>\ell \times n</math> matrix such that <math>A^* A = B^* B,</math>
then there is a <math>\ell \times k</math> matrix <math>Q</math> with orthonormal columns (meaning <math>Q^* Q = I_{k \times k}</math>) such that <math>B = Q A
When <math>\ell = k</math> this means <math>Q</math> is [[unitary matrix|unitary]].
This statement has an intuitive geometric interpretation in the real case:
let the columns of <math>A</math> and <math>B</math> be the vectors <math>a_1,\dots,a_n</math> and <math>b_1, \dots, b_n</math> in <math>\mathbb{R}^k
A real unitary matrix is an [[orthogonal matrix]], which describes a [[rigid transformation]] (an isometry of Euclidean space <math>\mathbb{R}^k</math>) preserving the 0 point (i.e. [[Rotation matrix|rotations]] and [[Reflection matrix|reflections]], without translations).
Therefore, the dot products <math>a_i \cdot a_j</math> and <math>b_i \cdot b_j</math> are equal if and only if some rigid transformation of <math>\mathbb{R}^k</math> transforms the vectors <math>a_1,\dots,a_n</math> to <math>b_1,\dots,b_n</math> (and 0 to 0).
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===Square root===
{{main|Square root of a matrix}}
A Hermitian matrix <math>M</math> is positive semidefinite if and only if there is a positive semidefinite matrix <math>B</math> (in particular <math>B</math> is Hermitian, so <math>B^* = B</math>) satisfying <math>M = B B
When <math>M</math> is positive definite, so is <math>M^\frac{1}{2},</math> hence it is also called the ''positive square root'' of <math>M
The non-negative square root should not be confused with other decompositions <math>M = B^*
Some authors use the name ''square root'' and <math>M^\frac{1}{2}</math> for any such decomposition, or specifically for the [[Cholesky decomposition]],
or any decomposition of the form <math>M = B B;</math>
others only use it for the non-negative square root.
If <math>M \succ N \succ 0</math> then <math>M^\frac{1}{2} \succ N^\frac{1}{2} \succ 0
===Cholesky decomposition===
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== Other characterizations ==
Let <math>M</math> be an <math>n \times n</math> [[Hermitian matrix|real symmetric matrix]], and let <math>B_1(M) \equiv \{ \mathbf{x} \in \mathbb{R}^n : \mathbf{x}^\top M\mathbf{x} \leq 1\}</math> be the "unit ball" defined by <math>M
* <math>B_1( \mathbf{v}\mathbf{v}^\top )</math> is a solid slab sandwiched between <math>\pm \{ \mathbf{w}: \langle \mathbf{w}, \mathbf{v}\rangle = 1 \}
* <math>M \succeq 0</math> if and only if <math>B_1(M)</math> is an ellipsoid, or an ellipsoidal cylinder.
* <math>M \succ 0</math> if and only if <math>B_1(M)</math> is bounded, that is, it is an ellipsoid.
* If <math>N \succ 0,</math> then <math>M \succeq N</math> if and only if <math>B_1(M) \subseteq B_1(N);</math> <math>M \succ N</math> if and only if <math>B_1(M) \subseteq \operatorname{int}
* If <math>N \succ 0,</math> then <math>M \succeq \frac{ \mathbf{v}\mathbf{v}^\top }{\mathbf{v}^\top N\mathbf{v}}</math> for all <math>v \neq 0</math> if and only if <math display="inline">B_1(M) \subset \bigcap_{ \mathbf{v}^\top N\mathbf{v} = 1 } B_1(\mathbf{v} \mathbf{v}^\top)
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 \mathbf{x}, \mathbf{y} \rangle \equiv \mathbf{y}^* M\mathbf{x}</math> for all <math>\mathbf{x}</math> and <math>\mathbf{y}</math> in <math>\mathbb{C}^n,</math> where <math>\mathbf{y}^*</math> is the conjugate transpose of <math>\mathbf{y}
; Its leading principal minors are all positive: The {{mvar|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 {{mvar|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.
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== Quadratic forms ==
{{Main|Definite quadratic form}}
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 \to \mathbb{R}</math> such that <math>Q(\mathbf{x}) = \mathbf{x}^\top M \mathbf{x}</math> for all <math>\mathbf{x}
A symmetric matrix <math>M</math> is positive definite if and only if its quadratic form is a [[strictly convex function]].
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One symmetric matrix and another matrix that is both symmetric and positive definite can be [[diagonalizable matrix#Simultaneous diagonalization|simultaneously diagonalized]]. This is so although simultaneous diagonalization is not necessarily performed with 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>\left(M - \lambda N\right)\mathbf{x} = 0</math> where we impose that <math>\mathbf{x}</math> be normalized, i.e. <math>\mathbf{x}^\top N \mathbf{x} = 1
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.
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== 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
===Inverse of positive definite matrix===
Every positive definite matrix is [[invertible matrix|invertible]] and its inverse is also positive definite.<ref>{{harvtxt|Horn|Johnson|2013}}, p. 438, Theorem 7.2.1</ref> If <math>M \geq N > 0</math> then <math>N^{-1} \geq M^{-1} > 0
=== Scaling ===
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===Multiplication===
* If <math>M</math> and <math>N</math> are positive definite, then the products <math>M N M</math> and <math>NMN</math> are also positive definite. If <math>M N = N M,</math> then <math>M N</math> is also positive definite.
* If <math>M</math> is positive semidefinite, then <math>A^* M A</math> is positive semidefinite for any (possibly rectangular) matrix <math>A
===Trace===
The diagonal entries <math>m_{ii}</math> of a positive-semidefinite matrix are real and non-negative. As a consequence the [[trace (linear algebra)|trace]], <math>\operatorname{tr}(M) \ge 0
<math display="block">\left|m_{ij}\right| \leq \sqrt{m_{ii}m_{jj}} \quad \forall i, j</math>
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An <math>n \times n</math> Hermitian matrix <math>M</math> is positive definite if it satisfies the following trace inequalities:<ref>{{cite journal | title=Bounds for Eigenvalues using Traces | last1=Wolkowicz | first1=Henry | last2 = Styan | first2 = George P.H. | journal=Linear Algebra and Its Applications | issue=29 | publisher=Elsevier | year=1980 | volume=29 | pages=471–506 | doi=10.1016/0024-3795(80)90258-X }}</ref>
<math display="block">\operatorname{tr}(M) > 0 \quad \mathrm{and} \quad \frac{(\operatorname{tr}(M))^2}{\operatorname{tr}(M^2)} > n-1
Another important result is that for any <math>M</math> and <math>N</math> positive-semidefinite matrices, <math>\operatorname{tr}(MN) \ge 0
===Hadamard product===
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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>\det(M \circ N) \geq \det(M) \det(N)
===Kronecker product===
If <math>M, N \geq 0,</math> although <math>M N</math> is not necessary positive semidefinite, the [[Kronecker product]] <math>M \otimes N \geq 0
===Frobenius product===
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===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 {{math|0}} and {{math|1}}, <math>\alpha M + \left(1 - \alpha\right) N</math> is also positive semidefinite. For any vector <math>\mathbf{x}</math>:
<math display="block">\mathbf{x}^\top \left(\alpha M + \left(1 - \alpha\right)N\right)\mathbf{x} = \alpha \mathbf{x}^\top M\mathbf{x} + (1 - \alpha) \mathbf{x}^\top N\mathbf{x} \geq 0
This property guarantees that [[semidefinite programming]] problems converge to a globally optimal solution.
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The positive-definiteness of a matrix <math>A</math> expresses that the angle <math>\theta</math> between any vector <math>\mathbf{x}</math> and its image <math>A \mathbf{x}</math> is always <math>-\pi / 2 < \theta < +\pi / 2:</math>
<math display="block">\cos\theta = \frac{ \mathbf{x}^\top A\mathbf{x} }{\lVert \mathbf{x} \rVert \lVert A\mathbf{x} \rVert} = \frac{\langle \mathbf{x}, A\mathbf{x} \rangle}{\lVert \mathbf{x} \rVert \lVert A\mathbf{x} \rVert} , \theta = \theta(\mathbf{x}, A \mathbf{x}) \equiv \widehat{\left(\mathbf{x},A\mathbf{x}\right)} \equiv</math> the angle between <math>\mathbf{x}</math> and <math>A\mathbf{x}
===Further properties===
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# 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 {{mvar|k}}th pivot during [[LU decomposition]].
# A matrix is negative definite if its {{mvar|k}}th order leading [[principal minor]] is negative when <math>k</math> is odd, and positive when <math>k</math> is even.
# If <math>M</math> is a real positive definite matrix, then there exists a positive real number <math>m</math> such that for every vector <math>\mathbf{v},</math> <math>\mathbf{v}^\top M\mathbf{v} \geq m\|\mathbf{v}\|_2^{
# 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 {{math|0}} and {{math|−1 .}}
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<math display="block">M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}</math>
where each block is <math>n \times n
We have that <math>\mathbf{z}^* M\mathbf{z} \ge 0</math> for all complex <math>\mathbf{z},</math> and in particular for <math>\mathbf{z} = [\mathbf{v}, 0]^\top .</math> Then
<math display="block">\begin{bmatrix} \mathbf{v}^* & 0 \end{bmatrix} \begin{bmatrix} A & B \\ B^* & D \end{bmatrix} \begin{bmatrix} \mathbf{v} \\ 0 \end{bmatrix} = \mathbf{v}^* A\mathbf{v} \ge 0
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. The argument can be extended to show that any [[Matrix_(mathematics)#Submatrix|principal submatrix]] of <math>M</math> is itself positive definite.
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=== Local extrema ===
A general [[quadratic form]] <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}^\top 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}
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.
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== 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>\mathcal{R_e} \left\{\mathbf{z}^* M \mathbf{z}\right\} > 0</math> for all non-zero complex vectors <math>\mathbf{z},</math> where <math>\mathcal{R_e}\{c\}</math> denotes the real part of a [[complex number]] <math>c
A non-symmetric real matrix with only positive eigenvalues may have a symmetric part with negative eigenvalues, in which case it will not be positive (semi)definite. For example, the matrix <math display=inline>M = \left[\begin{matrix} 4 & 9 \\ 1 & 4 \end{matrix}\right]</math> has positive eigenvalues 1 and 7, yet <math>\mathbf{x}^\top M \mathbf{x} = -2 </math> with the choice <math>\mathbf{x} = \left[\begin{smallmatrix} -1 \\ 1 \end{smallmatrix}\right] </math>.
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== Applications ==
=== Heat conductivity matrix ===
Fourier's law of heat conduction, giving heat flux <math>\mathbf{q}</math> in terms of the temperature gradient <math>\mathbf{g} = \nabla T</math> is written for anisotropic media as <math>\mathbf{q} = -K \mathbf{g},</math> in which <math>K</math> is the [[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>\mathbf{g}</math> always points from cold to hot, the heat flux <math>\mathbf{q}</math> is expected to have a negative inner product with <math>\mathbf{g}</math> so that <math>\mathbf{q}^\top \mathbf{g} < 0
More generally in thermodynamics, the flow of heat and particles is a fully coupled system as described by the [[Onsager reciprocal relations]], and the coupling matrix is required to be positive semi-definite (possibly non-symmetric) in order that entropy production be nonnegative.
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