Matrix congruence
Encyclopedia
In mathematics
, two matrices
A and B over a field
are called congruent if there exists an invertible matrix P over the same field such that
where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation
.
Matrix congruence arises when considering the effect of change of basis
on the Gram matrix attached to a bilinear form or quadratic form
on a finite-dimensional vector space
: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases
.
Note that Halmos
defines congruence in terms of conjugate transpose
(with respect to a complex inner product space
) rather than transpose, but this definition has not been adopted by most other authors.
states that two congruent symmetric matrices with real
entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, two matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
A and B over a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
are called congruent if there exists an invertible matrix P over the same field such that
- PTAP = B
where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
.
Matrix congruence arises when considering the effect of change of basis
Change of basis
In linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.-Expression of a basis:...
on the Gram matrix attached to a bilinear form or quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
on a finite-dimensional vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
.
Note that Halmos
Paul Halmos
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of probability theory, statistics, operator theory, ergodic theory, and functional analysis . He was also recognized as a great mathematical expositor.-Career:Halmos obtained his B.A...
defines congruence in terms of conjugate transpose
Conjugate transpose
In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...
(with respect to a complex inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors...
) rather than transpose, but this definition has not been adopted by most other authors.
Congruence over the reals
Sylvester's law of inertiaSylvester's law of inertia
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...
states that two congruent symmetric matrices with real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.