Sylvester's law of inertia

Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said to transform A into another symmetric matrix B = SAS^T, also of order n, where S^T is the transpose of S. It is also said that matrices A and B are congruent. If A is the coefficient matrix of some quadratic form of Rⁿ, then B is the matrix for the same form after the change of basis defined by S.

A symmetric matrix A can always be transformed in this way into a diagonal matrix D which has only entries 0, +1 and −1 along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of A, i.e. it does not depend on the matrix S used.

The number of +1s, denoted n₊, is called the positive index of inertia of A, and the number of −1s, denoted n₋, is called the negative index of inertia. The number of 0s, denoted n₀, is the dimension of the null space of A, known as the nullity of A. These numbers satisfy an obvious relation

${\displaystyle n_{0}+n_{+}+n_{-}=n.}$

The difference, sgn(A) = n₊ − n₋, is usually called the signature of A. (However, some authors use that term for the triple (n₀, n₊, n₋) consisting of the nullity and the positive and negative indices of inertia of A; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data.)

If the matrix A has the property that every principal upper left k × k minor Δ_k is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence

${\displaystyle \Delta _{0}=1,\Delta _{1},\ldots ,\Delta _{n}=\det A.}$

The law can also be stated as follows: two symmetric square matrices of the same size have the same number of positive, negative and zero eigenvalues if and only if they are congruent[3] (${\displaystyle B=SAS^{T}}$, for some non-singular ${\displaystyle S}$).

The positive and negative indices of a symmetric matrix A are also the number of positive and negative eigenvalues of A. Any symmetric real matrix A has an eigendecomposition of the form QEQ^T where E is a diagonal matrix containing the eigenvalues of A, and Q is an orthonormal square matrix containing the eigenvectors. The matrix E can be written E = WDW^T where D is diagonal with entries 0, +1, or −1, and W is diagonal with W_ii = √|E_ii|. The matrix S = QW transforms D to A.

In the context of quadratic forms, a real quadratic form Q in n variables (or on an n-dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from x to y) be brought to the diagonal form

${\displaystyle Q(x_{1},x_{2},\ldots ,x_{n})=\sum _{i=1}^{n}a_{i}x_{i}^{2}}$

with each a_i ∈ {0, 1, −1}. Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of Q, i.e., does not depend on a particular choice of diagonalizing basis. Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is positive definite (respectively, negative definite) have the same dimension. These dimensions are the positive and negative indices of inertia.

Sylvester's law of inertia is also valid if A and B have complex entries. In this case, it is said that A and B are *-congruent if and only if there exists a non-singular complex matrix S such that B = SAS^∗.

In the complex scenario, a way to state Sylvester's law of inertia is that if A and B are Hermitian matrices, then A and B are *-congruent if and only if they have the same inertia. A theorem due to Ikramov generalizes the law of inertia to any normal matrices A and B:[4]

If A and B are normal matrices, then A and B are congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.

• Metric signature
• Morse theory
• Cholesky decomposition
• Haynsworth inertia additivity formula

• Garling, D. J. H. (2011). Clifford algebras. An introduction. London Mathematical Society Student Texts. 78. Cambridge: Cambridge University Press. ISBN 978-1-107-09638-7. Zbl 1235.15025.

• Sylvester's law at PlanetMath.
• Sylvester's law of inertia and *-congruence