In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two
square matrices ''A'' and ''B'' over a
field are called congruent if there exists an
invertible matrix ''P'' over the same field such that
:''P''
T''AP'' = ''B''
where "T" denotes the
matrix transpose. Matrix congruence is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
.
Matrix congruence arises when considering the effect of
change of basis on the
Gram matrix attached to a
bilinear form
In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
or
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 ...
on a
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, ยง2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to disti ...
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.
Congruence over the reals
Sylvester's law of inertia 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.
See also
*
Congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
*
Matrix similarity
*
Matrix equivalence
References
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{{Matrix classes
Linear algebra
Matrices
Equivalence (mathematics)