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 th ...
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, ''U ...
about certain properties of the
coefficient matrix In linear algebra, a coefficient matrix is a matrix consisting of the coefficients of the variables in a set of linear equations. The matrix is used in solving systems of linear equations.
Coefficient matrix
In general, a system with ''m'' linear ...
of a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
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 a ...
that remain
invariant
Invariant and invariance may refer to:
Computer science
* Invariant (computer science), an expression whose value doesn't change during program execution
** Loop invariant, a property of a program loop that is true before (and after) each iteratio ...
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 considere ...
. 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 re ...
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 ro ...
who published its proof in 1852.
[
][
]
Statement
Let ''A'' be a symmetric square matrix of order ''n'' with
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
entries. Any
non-singular matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
''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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
. 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 ma ...
''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 a function, domain of the map which is mapped to the zero vector. That is, given a linear map between two vector space ...
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
Minor may refer to:
* Minor (law), a person under the age of certain legal activities.
** A person who has not reached the age of majority
* Academic minor, a secondary field of study in undergraduate education
Music theory
*Minor chord
** Barb ...
''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
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ...
(
, 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 denoted b ...
s of ''A''. Any symmetric real matrix ''A'' has an
eigendecomposition
In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matri ...
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 un ...
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 a ...
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 f ...
(respectively, negative definite) have the same
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
. 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 con ...
. In the complex scenario, a way to state Sylvester's law of inertia is that if ''A'' and ''B'' are
Hermitian matrices
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
, 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. A ...
''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. A ...
, 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
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative an ...
*
Morse theory
In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiabl ...
*
Cholesky decomposition
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced ) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for effici ...
*
Haynsworth inertia additivity formula
In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partit ...
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