In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector where is the row vector transpose of Print ed. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector where denotes the conjugate transpose of
Positive semi-definite matrices are defined similarly, except that the scalars and 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.
Some authors use more general definitions of definiteness, permitting the matrices to be non-symmetric or non-Hermitian. The properties of these generalized definite matrices are explored in , below, but are not the main focus of this article.
Definitions In the following definitions, is the transpose of is the conjugate transpose of and denotes the zero-vector.
Definitions for real matrices An symmetric real matrix is said to be positive-definite if for all non-zero in Formally,
An symmetric real matrix is said to be positive-semidefinite or non-negative-definite if for all in Formally,
An symmetric real matrix is said to be negative-definite if for all non-zero in Formally,
An symmetric real matrix is said to be negative-semidefinite or non-positive-definite if for all in Formally,
An 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 Notice that this is always a real number for any Hermitian square matrix
An Hermitian complex matrix is said to be positive-definite if for all non-zero in Formally,
An Hermitian complex matrix is said to be positive semi-definite or non-negative-definite if for all in Formally,
An Hermitian complex matrix is said to be negative-definite if for all non-zero in Formally,
An Hermitian complex matrix is said to be negative semi-definite or non-positive-definite if for all in Formally,
An 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 is positive-definite if and only if is real and positive for every non-zero complex column vectors This condition implies that is Hermitian (i.e. its transpose is equal to its conjugate), since being real, it equals its conjugate transpose for every which implies
By this definition, a positive-definite real matrix is Hermitian, hence symmetric; and is positive for all non-zero real column vectors However the last condition alone is not sufficient for to be positive-definite. For example, if
then for any real vector with entries and we have which is always positive if is not zero. However, if is the complex vector with entries and , one gets
which is not real. Therefore, is not positive-definite.
On the other hand, for a symmetric real matrix the condition " for all nonzero real vectors " does imply that is positive-definite in the complex sense.
Notation If a Hermitian matrix is positive semi-definite, one sometimes writes and if is positive-definite one writes To denote that is negative semi-definite one writes and to denote that is negative-definite one writes
The notation comes from functional analysis where positive semidefinite matrices define positive operators. If two matrices and satisfy we can define a non-strict partial order that is reflexive, antisymmetric, and transitive; it is not a total order, however, as in general, may be indefinite. Also note that and do not satisfy the usual correspondence of non-strict and strict partial order relations, because does not imply .
A common alternative notation is and for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.
Ramifications
It follows from the above definitions that a Hermitian matrix is positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a Hermitian matrix is positive-definite if and only if it defines an inner product.
Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A Hermitian matrix is positive-definite if and only if it satisfies any of the following equivalent conditions. is congruent with a diagonal matrix with positive real entries. is Hermitian, and all its eigenvalues are real and positive. is Hermitian, and all its leading principal minors are positive. There exists an invertible matrix with conjugate transpose such that 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, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point then the function is convex near , and, conversely, if the function is convex near then the Hessian matrix is positive-semidefinite at
The set of positive definite matrices is an open convex cone, while the set of positive semi-definite matrices is a closed convex cone.
Examples
Eigenvalues Let be an Hermitian matrix (this includes real symmetric matrices). All eigenvalues of are real, and their signs characterize its definiteness: is positive definite if and only if all of its eigenvalues are positive. is positive semi-definite if and only if all of its eigenvalues are non-negative. is negative definite if and only if all of its eigenvalues are negative. is negative semi-definite if and only if all of its eigenvalues are non-positive. is indefinite if and only if it has both positive and negative eigenvalues.
Let be an eigendecomposition of where is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of and is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix may be regarded as a diagonal matrix that has been re-expressed in coordinates of the (eigenvectors) basis Put differently, applying to some vector giving is the same as changing the basis to the eigenvector coordinate system using giving applying the stretching transformation to the result, giving and then changing the basis back using giving
With this in mind, the one-to-one change of variable shows that is real and positive for any complex vector if and only if is real and positive for any in other words, if is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal – that is, every eigenvalue of – 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 alternating signs when the characteristic polynomial of a real, symmetric matrix is available.
Decomposition
Let be an Hermitian matrix. is positive semidefinite if and only if it can be decomposed as a product
of a matrix with its conjugate transpose.
When is real, can be real as well and the decomposition can be written as
is positive definite if and only if such a decomposition exists with invertible. More generally, is positive semidefinite with rank if and only if a decomposition exists with a matrix of full row rank (i.e. of rank ). Moreover, for any decomposition , p. 440, Theorem 7.2.7
The columns of can be seen as vectors in the complex or real vector space respectively. Then the entries of are inner products (that is dot products, in the real case) of these vectors
In other words, a Hermitian matrix is positive semidefinite if and only if it is the Gram matrix of some vectors 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 equals the dimension of the space spanned by these vectors., p. 441, Theorem 7.2.10
Uniqueness up to unitary transformations The decomposition is not unique: if for some matrix and if is any unitary matrix (meaning ), then for
However, this is the only way in which two decompositions can differ: The decomposition is unique up to unitary transformations. More formally, if is a matrix and is a matrix such that then there is a matrix with orthonormal columns (meaning ) such that , p. 452, Theorem 7.3.11 When this means is unitary.
This statement has an intuitive geometric interpretation in the real case: let the columns of and be the vectors and in A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space ) preserving the 0 point (i.e. rotations and reflections, without translations). Therefore, the dot products and are equal if and only if some rigid transformation of transforms the vectors to (and 0 to 0).
Square root
A Hermitian matrix is positive semidefinite if and only if there is a positive semidefinite matrix (in particular is Hermitian, so ) satisfying This matrix is unique,, p. 439, Theorem 7.2.6 with is called the non-negative square root of and is denoted with When is positive definite, so is hence it is also called the positive square root of
The non-negative square root should not be confused with other decompositions Some authors use the name square root and for any such decomposition, or specifically for the Cholesky decomposition, or any decomposition of the form others only use it for the non-negative square root.
If then
Cholesky decomposition A Hermitian positive semidefinite matrix can be written as where is lower triangular with non-negative diagonal (equivalently where is upper triangular); this is the Cholesky decomposition. If is positive definite, then the diagonal of is positive and the Cholesky decomposition is unique. Conversely if is lower triangular with nonnegative diagonal then is positive semidefinite. The Cholesky decomposition is especially useful for efficient numerical calculations. A closely related decomposition is the LDL decomposition, where is diagonal and is lower unitriangular.
Williamson theorem Any positive definite Hermitian real matrix can be diagonalized via symplectic (real) matrices. More precisely, Williamson's theorem ensures the existence of symplectic and diagonal real positive such that .
Other characterizations Let be an real symmetric matrix, and let be the "unit ball" defined by Then we have the following is a solid slab sandwiched between if and only if is an ellipsoid, or an ellipsoidal cylinder. if and only if is bounded, that is, it is an ellipsoid. If then if and only if if and only if If then for all if and only if So, since the polar dual of an ellipsoid is also an ellipsoid with the same principal axes, with inverse lengths, we have That is, if is positive-definite, then for all if and only if
Let be an Hermitian matrix. The following properties are equivalent to being positive definite: