What Is the Inertia of a Matrix?

The inertia of a real symmetric n\times n matrix A is a triple, written \mathrm{In}(A) = (i_+(A),i_-(A),i_0(A)), where i_+(A) is the number of positive eigenvalues of A, i_-(A) is the number of negative eigenvalues of A, and i_0(A) is the number of zero eigenvalues of A.

The rank of A is i_+(A) + i_-(A). The difference i_+(A) - i_-(A) is called the signature.

In general it is not possible to determine the inertia by inspection, but some deductions can be made. If A has both positive and negative diagonal elements then i_+(A) > 1 and i_-(A) > 1. But in general the diagonal elements do not tell us much about the inertia. For example, here is a matrix that has positive diagonal elements but only one positive eigenvalue (and this example works for any n):

>> n = 4; A = -eye(n) + 2*ones(n), eigA = eig(sym(A))'
A =
     1     2     2     2
     2     1     2     2
     2     2     1     2
     2     2     2     1
eigA =
[-1, -1, -1, 7]

A congruence transformation of a symmetric matrix A is a transformation A \to X^T\!AX for a nonsingular matrix X. The result is clearly symmetric. Sylvester’s law of inertia (1852) says that the inertia is preserved under congruence transformations.

Theorem 1 (Sylvester’s law of inertia).

If A\in\mathbb{R}^{n\times n} is symmetric and X\in\mathbb{R}^{n\times n} is nonsingular then \mathrm{In}(A) = \mathrm{In}(X^T\!AX).

Sylvester’s law gives a way to determine the inertia without computing eigenvalues: find a congruence transformation that transforms A to a matrix whose inertia can be easily determined. A factorization PAP^T = LDL^T does the job, where P is a permutation matrix, L is unit lower triangular, and D is diagonal Then \mathrm{In}(A) = \mathrm{In}(D), and \mathrm{In}(D) can be read off the diagonal of D. This factorization does not always exist, and if it does exist is can be numerically unstable. A block LDL^T factorization, in which D is block diagonal with diagonal blocks of size 1 or 2, always exists, and its computation is numerically stable with a suitable pivoting strategy such as symmetric rook pivoting.

For the matrix above we can compute a block LDL^T factorization using the MATLAB ldl function:

>> [L,D,P] = ldl(A); D
D =
   1.0000e+00   2.0000e+00            0            0
   2.0000e+00   1.0000e+00            0            0
            0            0  -1.6667e+00            0
            0            0            0  -1.4000e+00

Since the leading 2-by-2 block of D has negative determinant and hence one positive eigenvalue and one negative eigenvalue, it follows that A has one positive eigenvalue and three negative eigenvalues.

A congruence transformation preserves the signs of the eigenvalues but not their magnitude. A result of Ostrowski (1959) bounds the ratios of the eigenvalues of the original and transformed matrices. Let the eigenvalues of a symmetric matrix be ordered \lambda_n \le \cdots \le \lambda_1.

Theorem (Ostrowski).

For a symmetric A\in \mathbb{R}^{n\times n} and X\in\mathbb{R}^{n\times n},

\lambda_k(X^*AX) = \theta_k \lambda_k(A), \quad k=1\colon n,

where \lambda_n(X^*X) \le \theta_k \le \lambda_1(X^*X).

The theorem shows that the further X is from being orthogonal the greater the potential change in the eigenvalues.

Finally, we note that everything here generalizes to complex Hermitian matrices by replacing transpose by conjugate transpose.

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This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.

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