Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of coordinates. Namely, if A is the symmetric matrix that defines the quadratic form, and S is a matrix of the same size such as D = SAST is diagonal, then the numbers of negative elements and of positive elements in the diagonal of D are always the same, for any such S.
This property is named for J. J. Sylvester who published its proof in 1852.[1][2]
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Statement of the theorem
Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size transforms A into another symmetric matrix B of order n defined by the rule
where ST is the transpose of S. 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 coordinates defined by S.
A symmetric matrix A can always be transformed into an equivalent 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 kernel of A, and also the corank of A. These numbers satisfy an obvious relation
The difference sign(A) = n− − n+) is usually called the signature of A. (However, some authors use that term for the whole triple (n0, n+, n−) consisting of the corank and the positive and negative indices of inertia of A.)
If the matrix A has the property that every principal upper left k×k minor Δk is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence
The positive and negative indices of inertia of A can also be characterized as the numbers of positive and negative eigenvalues of A, but this characterization is harder to use in practice.
Law of inertia for quadratic forms
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 be brought to the diagonal form
with each ai ∈ {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.
See also
References
- ^ Sylvester, J J (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares". Philosophical Magazine IV: 138–142. http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf. Retrieved 2008-06-27.
- ^ Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 0-19-853248-2.
External links
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