Definiteness of A Matrix
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Definiteness of A Matrix

In linear algebra, a symmetric real matrix is said to be positive-definite if the scalar is strictly positive for every non-zero column vector of real numbers. Here denotes the transpose of .[1] When interpreting as the output of an operator, , that is acting on an input, , the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. Put differently, that applying M to z (Mz) keeps the output in the direction of z.

More generally, a complex Hermitian matrix is said to be positive-definite if the scalar is strictly positive for every non-zero column vector of complex numbers. Here denotes the conjugate transpose of . Note that is automatically real since is Hermitian.

Positive semi-definite matrices are defined similarly, except that the above scalars or 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 is positive-definite if and only if the bilinear form is 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.[2]

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

Definitions

In the following definitions, is the transpose of , is the conjugate transpose of and denotes the n-dimensional 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 all non-zero complex column vectors ". This condition implies that is Hermitian (i.e. its transpose is equal to its conjugate). To see this, consider the matrices and , so that and . The matrices and are Hermitian, therefore and are individually real. If is real, then must be zero for all . Then is the zero matrix and , proving that is Hermitian.

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 notion comes from functional analysis where positive semidefinite matrices define positive operators.

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.

Examples

  • The identity matrix 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
    .

    Seen as a complex matrix, for any non-zero column vector z with complex entries a and b one has

    .
    Either way, the result is positive since is not the zero vector (that is, at least one of and is not zero).
  • The real symmetric matrix
    is positive-definite since for any non-zero column vector z with entries a, b and c, we have
    This result is a sum of squares, and therefore non-negative; and is zero only if , that is, when z is the zero vector.
  • For any real invertible matrix , the product is a positive definite matrix (if the means of the columns of A are 0, then this is also called the covariance matrix). A simple proof is that for any non-zero vector , the condition since the invertibility of matrix means that
  • The example 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
    for which

Eigenvalues

Let be an Hermitian matrix. This implies all its eigenvalues are real.

  • 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 (eigen vectors) basis . Put differently, applying M to some vector z in our coordinates system (Mz), is the same as changing the basis of our z to the eigen vector coordinate system using P-1 (P-1z), applying the stretching transformation D to it (DP-1z), and then changing the basis back to our system using P (PDP-1z).

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 , .[3]

Proof

If , then , so is positive semidefinite. If moreover is invertible then the inequality is strict for , so is positive definite. If is of rank , then .

In the other direction, suppose is positive semidefinite. Since is Hermitian, it has an eigendecomposition where is unitary and is a diagonal matrix whose entries are the eigenvalues of Since is positive semidefinite, the eigenvalues are non-negative real numbers, so one can define as the diagonal matrix whose entries are non-negative square roots of eigenvalues. Then for . If moreover is positive definite, then the eigenvalues are (strictly) positive, so is invertible, and hence is invertible as well. If has rank , then it has exactly positive eigenvalues and the others are zero, hence in all but rows are all zeroed. Cutting the zero rows gives a matrix such that .

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.[4]

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 .[5] 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 matrix is positive semidefinite if and only if there is a positive semidefinite matrix (in particular is Hermitian, so ) satisfying . This matrix is unique,[6] 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 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 ; other only use it for the non-negative square root.

If then .

Cholesky decomposition

A 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. 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.

Other characterizations

Let be an Hermitian matrix. The following properties are equivalent to being positive definite:

The associated sesquilinear form is an inner product
The sesquilinear form defined by is the function from to such that for all and in , where is the conjugate transpose of . For any complex matrix , this form is linear in and semilinear in . Therefore, the form is an inner product on if and only if is real and positive for all nonzero ; that is if and only if is positive definite. (In fact, every inner product on arises in this fashion from a Hermitian positive definite matrix.)
Its leading principal minors are all positive
The kth leading principal minor of a matrix is the determinant of its upper-left 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 kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row , Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. This condition can be checked each time a new row of the triangular matrix is obtained.

A positive semidefinite matrix is positive definite if and only if it is invertible.[7] A matrix is negative (semi)definite if and only if is positive (semi)definite.

Quadratic forms

The (purely) quadratic form associated with a real matrix is the function such that for all . can be assumed symmetric by replacing it with .

A symmetric matrix is positive definite if and only if its quadratic form is a strictly convex function.

More generally, any quadratic function from to can be written as where is a symmetric matrix, is a real -vector, and a real constant. This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if is positive definite. For this reason, positive definite matrices play an important role in optimization problems.

Simultaneous diagonalization

A symmetric matrix and another symmetric and positive definite matrix can be simultaneously diagonalized, although not necessarily via 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.

Let be a symmetric and a symmetric and positive definite matrix. Write the generalized eigenvalue equation as where we impose that be normalized, i.e. . Now we use Cholesky decomposition to write the inverse of as . Multiplying by and letting , we get , which can be rewritten as where . Manipulation now yields where is a matrix having as columns the generalized eigenvectors and is a diagonal matrix of the generalized eigenvalues. Now premultiplication with gives the final result: and , but note that this is no longer an orthogonal diagonalization with respect to the inner product where . In fact, we diagonalized with respect to the inner product induced by .[8]

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

For arbitrary square matrices , we write if i.e., is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict 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.[9] If then .[10] Moreover, by the min-max theorem, the kth largest eigenvalue of is greater than the kth largest eigenvalue of .

Scaling

If is positive definite and is a real number, then is positive definite.[11]

Addition

If and are positive definite, then the sum is also positive definite.[11]

Multiplication

  • If and are positive definite, then the products and are also positive definite. If , then is also positive definite.
  • If is positive semidefinite, then is positive semidefinite for any (possibly rectangular) matrix . If is positive definite and has full column rank, then is positive definite.[12]

Submatrices

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

Trace

The diagonal entries of a positive-semidefinite matrix are real and non-negative. As a consequence the trace, . Furthermore,[13] since every principal sub-matrix (in particular, 2-by-2) is positive semidefinite,

and thus, when ,

An Hermitian matrix is positive definite if it satisfies the following trace inequalities:[14]

Hadamard product

If , although is not necessary positive semidefinite, the Hadamard product is, (this result is often called the Schur product theorem).[15]

Regarding the Hadamard product of two positive semidefinite matrices , , there are two notable inequalities:

  • Oppenheim's inequality: [16]
  • .[17]

Kronecker product

If , although is not necessary positive semidefinite, the Kronecker product .

Frobenius product

If , although is not necessary positive semidefinite, the Frobenius product (Lancaster-Tismenetsky, The Theory of Matrices, p. 218).

Convexity

The set of positive semidefinite symmetric matrices is convex. That is, if and are positive semidefinite, then for any between 0 and 1, is also positive semidefinite. For any vector :

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

Relation with cosine

The positive-definiteness of a matrix expresses that the angle between any vector and its image is always :

Further properties

  1. If is a symmetric Toeplitz matrix, i.e. the entries are given as a function of their absolute index differences: , and the strict inequality

    holds, then is strictly positive definite.
  2. Let and Hermitian. If (resp., ) then (resp., ).[18]
  3. If is real, then there is a such that , where is the identity matrix.
  4. If denotes the leading minor, is the kth pivot during LU decomposition.
  5. A matrix is negative definite if its k-th order leading principal minor is negative when is odd, and positive when 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 0 and -1.

Block matrices

A positive matrix may also be defined by blocks:

where each block is . By applying the positivity condition, it immediately follows that and are hermitian, and .

We have that for all complex , and in particular for . Then

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

Converse results can be proved with stronger conditions on the blocks, for instance using the Schur complement.

Local extrema

A general quadratic form on real variables can always be written as where is the column vector with those variables, and is a symmetric real matrix. Therefore, the matrix being positive definite means that has a unique minimum (zero) when is zero, and is strictly positive for any other .

More generally, a twice-differentiable real function on real variables has local minimum at arguments 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

The definition of positive definite can be generalized by designating any complex matrix (e.g. real non-symmetric) as positive definite if for all non-zero complex vectors , where denotes the real part of a complex number .[19] Only the Hermitian part determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, If and are real, we have for all real nonzero vectors if and only if the symmetric part is positive definite in the narrower sense. It is immediately clear that 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 has positive eigenvalues yet is not positive definite; in particular a negative value of is obtained with the choice (which is the eigenvector associated with the negative eigenvalue of the symmetric part of ).

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

Fourier's law of heat conduction, giving heat flux in terms of the temperature gradient is written for anisotropic media as , in which 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 always points from cold to hot, the heat flux is expected to have a negative inner product with so that . Substituting Fourier's law then gives this expectation as , implying that the conductivity matrix should be positive definite.

See also

Notes

  1. ^ "Appendix C: Positive Semidefinite and Positive Definite Matrices". Parameter Estimation for Scientists and Engineers: 259-263. doi:10.1002/9780470173862.app3.
  2. ^ Stewart, J. (1976). "Positive definite functions and generalizations, an historical survey". Rocky Mountain J. Math. 6 (3): 409-434. doi:10.1216/RMJ-1976-6-3-409.
  3. ^ Horn & Johnson (2013), p. 440, Theorem 7.2.7
  4. ^ Horn & Johnson (2013), p. 441, Theorem 7.2.10
  5. ^ Horn & Johnson (2013), p. 452, Theorem 7.3.11
  6. ^ Horn & Johnson (2013), p. 439, Theorem 7.2.6 with
  7. ^ Horn & Johnson (2013), p. 431, Corollary 7.1.7
  8. ^ Horn & Johnson (2013), p. 485, Theorem 7.6.1
  9. ^ Horn & Johnson (2013), p. 438, Theorem 7.2.1
  10. ^ Horn & Johnson (2013), p. 495, Corollary 7.7.4(a)
  11. ^ a b Horn & Johnson (2013), p. 430, Observation 7.1.3
  12. ^ Horn & Johnson (2013), p. 431, Observation 7.1.8
  13. ^ Horn & Johnson (2013), p. 430
  14. ^ Wolkowicz, Henry; Styan, George P.H. (1980). "Bounds for Eigenvalues using Traces". Linear Algebra and its Applications. Elsevier (29): 471-506.
  15. ^ Horn & Johnson (2013), p. 479, Theorem 7.5.3
  16. ^ Horn & Johnson (2013), p. 509, Theorem 7.8.16
  17. ^ Styan, G. P. (1973). "Hadamard products and multivariate statistical analysis". Linear Algebra and Its Applications. 6: 217-240., Corollary 3.6, p. 227
  18. ^ Bhatia, Rajendra (2007). Positive Definite Matrices. Princeton, New Jersey: Princeton University Press. p. 8. ISBN 978-0-691-12918-1.
  19. ^ Weisstein, Eric W. Positive Definite Matrix. From MathWorld--A Wolfram Web Resource. Accessed on 2012-07-26

References

External links


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