Sylvester's law of inertia is a
theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
in
matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, '' ...
about certain properties of the
coefficient matrix of a
real quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
that remain
invariant under a
change of basis
In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
. Namely, if ''A'' is the
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
that defines the quadratic form, and ''S'' is any invertible matrix such that ''D'' = ''SAS''
T is diagonal, then the number of negative elements in the diagonal of ''D'' is always the same, for all such ''S''; and the same goes for the number of positive elements.
This property is named after
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...
who published its proof in 1852.
[
][
]
Statement
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 , 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 R
''n'', 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
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
''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 ''n''
0, is the dimension of the
null space
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kern ...
of ''A'', known as the nullity of ''A''. These numbers satisfy an obvious relation
:
The difference, , is usually called the signature of ''A''. (However, some authors use that term for the triple 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
minor ''Δ''
''k'' is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence
:
Statement in terms of eigenvalues
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 (
, for some non-singular
).
The positive and negative indices of a symmetric matrix ''A'' are also the number of positive and negative
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s 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
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
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''.
Law of inertia for quadratic forms
In the context of
quadratic form
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
s, 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
:
with each ''a''
''i'' ∈ . 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 In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite fu ...
(respectively, negative definite) have the same
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. These dimensions are the positive and negative indices of inertia.
Generalizations
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 , where * denotes the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
. 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, the definition of which is still valid as the eigenvalues of Hermitian matrices are always real numbers.
Ostrowski proved a quantitative generalization of Sylvester's law of inertia: if ''A'' and ''B'' are *-congruent with , then their eigenvalues ''λ''
i are related by
where ''θ
i'' are such that ''λ
n''(''SS
*'') ≤ ''θ
i'' ≤ ''λ
1''(''SS
*'').
A theorem due to Ikramov generalizes the law of inertia to any
normal matrices In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
''A'' and ''B'':
If ''A'' and ''B'' are
normal matrices In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
, 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.
See also
*
Metric signature
*
Morse theory
*
Cholesky decomposition
*
Haynsworth inertia additivity formula
References
*
External links
* {{PlanetMath , urlname=SylvestersLaw , title=Sylvester's law
Sylvester's law of inertia and *-congruence
Matrix theory
Quadratic forms
Theorems in linear algebra