Positive Definite Matrix

An n×n complex matrix A is called positive definite if


for all nonzero complex vectors x in C^n, where x^* denotes the conjugate transpose of the vector x. In the case of a real matrix A, equation (1) reduces to


where x^(T) denotes the transpose. Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models (Johnson 1970).

A matrix m may be tested to determine if it is positive definite in the Wolfram Language using PositiveDefiniteMatrixQ[m].

A linear system of equations with a positive definite matrix can be efficiently solved using the so-called Cholesky decomposition. A positive definite matrix has at least one matrix square root. Furthermore, exactly one of its matrix square roots is itself positive definite.

A necessary and sufficient condition for a complex matrix A to be positive definite is that the Hermitian part


where A^(H) denotes the conjugate transpose, be positive definite. This means that a real matrix A is positive definite iff the symmetric part


where A^(T) is the transpose, is positive definite (Johnson 1970).

Confusingly, the discussion of positive definite matrices is often restricted to only Hermitian matrices, or symmetric matrices in the case of real matrices (Pease 1965, Johnson 1970, Marcus and Minc 1988, p. 182; Marcus and Minc 1992, p. 69; Golub and Van Loan 1996, p. 140). A Hermitian (or symmetric) matrix is positive definite iff all its eigenvalues are positive. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues.

The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular.

If A and B are positive definite, then so is A+B. The matrix inverse of a positive definite matrix is also positive definite.

The definition of positive definiteness is equivalent to the requirement that the determinants associated with all upper-left submatrices are positive.

The following are necessary (but not sufficient) conditions for a Hermitian matrix A (which by definition has real diagonal elements a_(ii)) to be positive definite.

1. a_(ii)>0 for all i,

2. a_(ii)+a_(jj)>2|R[a_(ij)]| for i!=j,

3. The element with largest modulus lies on the main diagonal,

4. det(A)>0.

Here, R[z] is the real part of z, and a typo in Gradshteyn and Ryzhik (2000, p. 1063) has been corrected in item (ii).

A real symmetric matrix A is positive definite iff there exists a real nonsingular matrix M such that


where M^(T) is the transpose (Ayres 1962, p. 134). In particular, a 2×2 symmetric matrix

 [a b; b c]

is positive definite if


for all v=(v_1,v_2)!=0.

The numbers of positive definite n×n matrices of given types are summarized in the following table. For example, the three positive definite 2×2 (0,1)-matrices are

 [1 0; 0 1],[1 0; 1 1],[1 1; 0 1],

all of which have eigenvalue 1 with degeneracy of two.

matrix typeOEIScounts
(0,1)-matrixA0856561, 3, 27, 681, 43369, ...
(-1,1)-matrixA0061251, 2, 8, 64, 1024, ...
(-1,0,1)-matrixA0862151, 7, 311, 79505, ...

See also

Determinant, Eigenvalue, Hermitian Matrix, Matrix, Negative Definite Matrix, Negative Semidefinite Matrix, Positive Eigenvalued Matrix, Positive Matrix, Positive Semidefinite Matrix, Sylvester's Criterion

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Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, p. 134, 1962.Golub, G. H. and Van Loan, C. F. "Positive Definite Systems." §4.2 in Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins University Press, pp. 140-141, 1996.Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1106, 2000.Johnson, C. R. "Positive Definite Matrices." Amer. Math. Monthly 77, 259-264 1970.Lindell, I. V. Methods for Electromagnetic Field Analysis. New York: Clarendon Press, 1992.Marcus, M. and Minc, H. Introduction to Linear Algebra. New York: Dover, p. 182, 1988.Marcus, M. and Minc, H. "Positive Definite Matrices." §4.12 in A Survey of Matrix Theory and Matrix Inequalities. New York: Dover, p. 69, 1992.Pease, M. C. Methods of Matrix Algebra. New York: Academic Press, 1965.Sloane, N. J. A. Sequences A006125, A085656, and A086215 in "The On-Line Encyclopedia of Integer Sequences."

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Positive Definite Matrix

Cite this as:

Weisstein, Eric W. "Positive Definite Matrix." From MathWorld--A Wolfram Web Resource.

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