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

R[x^*Ax]>0

(1)

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

x^(T)Ax>0,

(2)

where x^(T) denotes the transpose. Positive definite matrices are of both theoretical and computational importance during a big variety of applications. they’re used, for instance , in optimization algorithms and within the construction of varied rectilinear regression models (Johnson 1970).

A linear system of equations with a positive definite matrix are often efficiently solved using the so-called Cholesky decomposition. A positive definite matrix has a minimum of one matrix root . Furthermore, exactly one among its matrix square roots is itself positive definite.

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

A_H=1/2(A+A^(H)),

(3)

where A^(H) denotes the conjugate transpose, be positive definite. this suggests that a true matrix A is positive definite iff the symmetric part

A_S=1/2(A+A^(T)),

(4)

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

Confusingly, the discussion of positive definite matrices is usually restricted to only Hermitian matrices, or symmetric matrices within 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 usually positive, so a positive definite matrix is usually nonsingular.

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

The definition of positive definiteness is like the need that the determinants related to 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 most diagonal,

4. det(A)>0.

Here, R[z] is that the real a 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 true square matrix M such

A=MM^(T),

(5)

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

[a b; b c]

(6)

is positive definite if

av_1^2+2bv_1v_2+cv_2^2>0

(7)

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],

(8)

all of which have eigenvalue 1 with degeneracy of two.

matrix type OEIS counts

(0,1)-matrix A085656 1, 3, 27, 681, 43369, …

(-1,0,1)-matrix A086215 1, 7, 311, 79505, …