Wikipedia / Definite matrix
Wikipedia / Definite matrix
In mathematics, a symmetric matrix with real entries is positive-definite is positive for every nonzero real column vector is the row vector transpose More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite
Positive semi-definite 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
is the conjugate transpose denotes the zero-vector.
Definitions for real matrices
is said to be positive-definite
is said to be positive-semidefinite or non-negative-definite
is said to be negative-definite
is said to be negative-semidefinite or non-positive-definite
symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.
Definitions for complex matrices
is said to be positive-definite Formally,
is said to be positive semi-definite or non-negative-definite Formally,
is said to be negative-definite Formally,
is said to be negative semi-definite or non-positive-definite Formally,
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.
By this definition, a positive-definite real is positive for all non-zero real
is the complex vector with entries and , one gets
is not positive-definite.
On the other hand, for a symmetric" does is positive-definite in the complex sense.
Notation
The notation comes from functional analysis where positive semidefinite matrices define positive operators we can define a non-strict partial order that is reflexive, antisymmetric, and transitive; it is not a total order do not satisfy the usual correspondence of non-strict and strict partial order relations.
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
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 then the function is convex near
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
Hermitian matrix (this includes real symmetric matrices 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.
be an eigendecomposition is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues is the same as changing the basis applying the stretching transformation
– 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 is available.
Decomposition
Hermitian matrix with its conjugate transpose.
can be seen as vectors in the complex or real vector space are inner products (that is dot products is positive semidefinite if and only if it is the Gram matrix It is positive definite if and only if it is the Gram matrix of some linearly independent equals the dimension of the space spanned by these vectors.
Uniqueness up to unitary transformations
is any unitary
However, this is the only way in which two decompositions can differ: The decomposition is unique up to unitary transformations is unitary.
A real unitary matrix is an orthogonal matrix, which describes a rigid transformation) preserving the 0 point (i.e. rotations and reflections (and 0 to 0).
Square root
is unique, is called the non-negative square root hence it is also called the positive square root
Some authors use the name square root for any such decomposition, or specifically for the Cholesky decomposition others only use it for the non-negative square root.
Cholesky decomposition
is upper triangular); this is the Cholesky decomposition is positive semidefinite. The Cholesky decomposition is especially useful for efficient numerical calculations. A closely related decomposition is the LDL decomposition is lower unitriangular.
Williamson theorem
can be diagonalized via symplectic (real) matrices. More precisely, Williamson's theorem.
Other characterizations
real symmetric matrix Then we have the following
is an ellipsoid, or an ellipsoidal cylinder.
is bounded, that is, it is an ellipsoid.
Hermitian matrix being positive definite:
The associated sesquilinear form is an inner product : The sesquilinear form Therefore, the form is an inner product arises in this fashion from a Hermitian positive definite matrix.)
Its leading principal minors are all positive : The th leading principal minor is the determinant 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 pivoting process. Since the of the triangular matrix is obtained.
A positive semidefinite matrix is positive definite if and only if it is invertible. is positive (semi)definite.
Quadratic forms
The (purely) quadratic form since any asymmetric part will be zeroed-out in the double-sided product.
is positive definite if and only if its quadratic form is a strictly convex function.
More generally, any quadratic function is a real case, we have
Theorem: is positive definite.
Proof: is a line or a downward parabola, thus not strictly convex and not having a global minimum.
For this reason, positive definite matrices play an important role in optimization problems.
Simultaneous diagonalization
One symmetric matrix and another matrix that is both symmetric and positive definite can be simultaneously diagonalized. This is so although simultaneous diagonalization is not necessarily performed with a 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.
Now we use Cholesky decomposition
Note that this result does not contradict what is said on simultaneous diagonalization in the article 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.
Properties
Induced partial ordering
is positive semi-definite. This defines a partial ordering The ordering is called the Loewner order.
Inverse of positive definite matrix
Every positive definite matrix is invertible and its inverse is also positive definite. Moreover, by the min-max theorem, the is greater than or equal to the
Scaling
is positive definite.
Addition
is also positive-definite.
is also positive-semidefinite.
is also positive-definite.
Multiplication
is also positive definite.
is positive definite.
Trace
of a positive-semidefinite matrix are real and non-negative. As a consequence the trace Furthermore,
is positive definite if it satisfies the following trace inequalities:
is positive-semidefinite and thus has non-negative eigenvalues, whose sum, the trace, is therefore also non-negative.
Hadamard product
is not necessary positive semidefinite, the Hadamard product (this result is often called the Schur product theorem).
there are two notable inequalities:
Kronecker product
is not necessary positive semidefinite, the Kronecker product
Frobenius product
is not necessary positive semidefinite, the Frobenius inner product (Lancaster–Tismenetsky, The Theory of Matrices, p.218).
Convexity
The set of positive semidefinite symmetric matrices is convex between and
This property guarantees that semidefinite programming problems converge to a globally optimal solution.
Relation with cosine
Further properties
is a symmetric Toeplitz matrix and the strict is strictly positive definite.
).
is the identity matrix.
is the th pivot during LU decomposition.
A matrix is negative definite if its th order leading principal minor is even.
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 and
Block matrices and submatrices
matrix may also be defined by blocks
must be positive definite. The argument can be extended to show that any principal submatrix is itself positive definite.
Converse results can be proved with stronger conditions on the blocks, for instance, using the Schur complement.
Local extrema
A general quadratic form
if its gradient is zero and its 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.
Covariance
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.
Extension for non-Hermitian square matrices
denotes the real part of a complex number
.
In summary, the distinguishing feature between the real and complex case is that, a 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
is the thermal conductivity should be symmetric, however it becomes nonsymmetric in the presence of a magnetic field as in a thermal Hall effect.
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.
See also
References
Sources
External links