Positive Definite Matrix

An complex matrix is called positive definite if

(1)

for all nonzero complex vectors , where denotes the conjugate transpose of the vector . In the case of a real matrix , equation (1) reduces to

(2)

where 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 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 to be positive definite is that the Hermitian part

(3)

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

(4)

where 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 and are positive definite, then so is . 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 (which by definition has real diagonal elements ) to be positive definite.

1. for all ,

2. for ,

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

4. .

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

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

(5)

where is the transpose (Ayres 1962, p. 134). In particular, a symmetric matrix

(6)

is positive definite if

(7)

for all .

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

(8)

all of which have eigenvalue 1 with degeneracy of two.