Matrix congruence
Matrix congruence is an equivalence relation on matrices that expresses the effect of applying the same change of coordinates to both arguments of a bilinear form. For square matrices (A,B\in M_n(F)) over a field (F), the matrices are congruent when an invertible matrix (P\in \operatorname{GL}_n(F)) satisfies
[ B=P^{\mathsf T}AP. ]
The notation (A\cong B) is commonly used when the intended relation is clear from context. Over fields equipped with an involution, especially the complex numbers, the corresponding relation often uses the conjugate transpose:
[ B=P^*AP. ]
This second relation is called (*)-congruence. Although congruence resembles matrix similarity, the two relations encode different mathematical objects. Similarity represents a change of basis for a linear operator and has the form (P^{-1}AP), whereas congruence represents a change of basis for a form and applies the transformation to both variables.
Interpretation through bilinear forms
Every matrix (A\in M_n(F)) determines a bilinear form on (F^n) by
[ \beta_A(x,y)=x^{\mathsf T}Ay. ]
Under the coordinate substitution (x=Pu) and (y=Pv), the same form is represented in the new basis by
[ \beta_A(Pu,Pv)=u^{\mathsf T}(P^{\mathsf T}AP)v. ]
Consequently, two matrices are congruent precisely when they represent the same bilinear form in two different bases. This interpretation also explains why congruence uses a transpose rather than an inverse on the left: the matrix acts on two coordinate arguments rather than representing an endomorphism of a single coordinate space.
Congruence is reflexive because (A=I^{\mathsf T}AI), symmetric because the defining equation can be inverted, and transitive because successive basis changes multiply. If (B=P^{\mathsf T}AP) and (C=Q^{\mathsf T}BQ), then
[ C=(PQ)^{\mathsf T}A(PQ). ]
The equivalence classes are therefore the isomorphism classes of bilinear forms on a fixed finite-dimensional vector space.
For a sesquilinear form over (\mathbb C), the coordinate expression is (x^*Ay), and a basis change produces (P^*AP). Hermitian matrices are preserved by this transformation because
[ (P^AP)^=P^*A^*P. ]
Thus (*)-congruence, rather than ordinary transpose congruence, is the natural relation for Hermitian forms.
Congruence invariants
Several properties remain unchanged throughout a congruence class. Since multiplication by invertible matrices does not alter matrix rank,
[ \operatorname{rank}(P^{\mathsf T}AP)=\operatorname{rank}(A). ]
The dimensions of the left and right radicals of the associated bilinear form are likewise preserved. For square matrices these dimensions agree with the nullity, although the two radical subspaces need not coincide for a general nonsymmetric form.
The determinant transforms according to
[ \det(P^{\mathsf T}AP)=\det(P)^2\det(A). ]
Accordingly, over a general field the determinant of a nonsingular matrix is preserved only up to multiplication by a nonzero square. This square class is an invariant of nonsingular symmetric bilinear forms. Under complex (*)-congruence, the corresponding factor is (\overline{\det(P)}\det(P)=|\det(P)|^2), which is a positive real number.
Structural identities are also preserved. If (A) is symmetric, then (P^{\mathsf T}AP) is symmetric; if (A) is alternating or skew-symmetric, its congruent transforms have the same property. These observations divide the classification problem into substantially different cases rather than providing a single canonical form for every square matrix.
Similarity invariants do not generally survive congruence. Eigenvalues, the characteristic polynomial, and the Jordan form can all change because congruence is not conjugation. For example, the real (1\times1) matrices ([1]) and ([4]) are congruent through (P=[2]), even though their eigenvalues are different.
Symmetric and Hermitian classification
When the characteristic of (F) is not (2), every symmetric matrix is congruent to a diagonal matrix. This statement follows from the decomposition of a symmetric bilinear form into mutually orthogonal one-dimensional components, with an additional zero block when the form is degenerate. Over an arbitrary field, the resulting diagonal coefficients retain arithmetic information because distinct elements need not belong to the same square class.
Over the real numbers, the classification is governed by Sylvester's law of inertia. Every real symmetric matrix is congruent to
[ \operatorname{diag}(I_p,-I_q,0_r), ]
where (p) is the number of positive directions, (q) is the number of negative directions, and (r) is the nullity. James Joseph Sylvester established that the triple ((p,q,r)) is independent of the diagonalization and therefore completely determines the real congruence class. The difference (p-q) is the signature, while (p+q) is the rank.
Every complex symmetric matrix of rank (r) is congruent over (\mathbb C) to
[ \operatorname{diag}(I_r,0). ]
The signs appearing in the real classification disappear because every nonzero complex number has a square root. This statement concerns transpose congruence of complex symmetric matrices and is distinct from unitary diagonalization or from the spectral theory of Hermitian matrices.
For a complex Hermitian matrix under ()-congruence, inertia again provides the complete classification. Such a matrix is ()-congruent to (\operatorname{diag}(I_p,-I_q,0_r)), and the three block dimensions are invariant. The result is the Hermitian analogue of the real law of inertia.
Alternating forms
An alternating bilinear form satisfies (\beta(x,x)=0) for every vector (x). Over a field of characteristic other than (2), its representing matrix is skew-symmetric, so (A^{\mathsf T}=-A). Every such matrix is congruent to a direct sum of blocks
[ J= \begin{pmatrix} 0&1\ -1&0 \end{pmatrix} ]
together with a zero block. The number of (J)-blocks is half the rank and completely determines the congruence class. In particular, an alternating matrix has even rank.
A nondegenerate alternating form exists only in even dimension and is congruent to the standard symplectic form. Its automorphism group consists of the invertible matrices (P) satisfying (P^{\mathsf T}JP=J), which form the symplectic group. In characteristic (2), alternating and skew-symmetric conditions no longer coincide in the same manner, so the diagonal behavior of the form must be retained explicitly.
General square matrices
The congruence classification of unrestricted square matrices is more involved than the symmetric or alternating cases. A nonsymmetric matrix represents a bilinear form without an interchange symmetry, and diagonalization by congruence is generally impossible. Canonical descriptions use direct sums of blocks associated with singular structure and with polynomial data derived from the nonsingular part.
For an invertible matrix (A), the matrix
[ A^{-\mathsf T}A ]
is called a cosquare. If (B=P^{\mathsf T}AP), then
[ B^{-\mathsf T}B=P^{-1}(A^{-\mathsf T}A)P, ]
so congruent matrices have similar cosquares. The similarity class of the cosquare is therefore a congruence invariant, although the reconstruction of a congruence class from this invariant requires compatibility conditions and additional data. The corresponding theory connects congruence with matrix pencils, elementary divisors, and the canonical decomposition of bilinear forms.
In the late nineteenth century, Leopold Kronecker incorporated related singular structures into his analysis of bilinear pencils. The resulting distinction between regular and singular components became part of the later canonical theory of matrix equivalence and congruence.
Quadratic forms
A matrix (A) also determines a quadratic form by
[ q_A(x)=x^{\mathsf T}Ax. ]
When the characteristic is not (2), the quadratic form depends only on the symmetric part
[ \frac{A+A^{\mathsf T}}{2}, ]
because the skew-symmetric part contributes zero to (x^{\mathsf T}Ax). Congruence of symmetric matrices then coincides with equivalence of the associated quadratic forms under invertible linear substitutions.
The relationship changes in characteristic (2), where division by (2) is unavailable and the polar form does not determine every quadratic form uniquely. Matrix congruence continues to classify bilinear forms, while the classification of quadratic forms requires additional information concerning diagonal terms and their behavior under coordinate changes. The Witt decomposition organizes a nonsingular quadratic space into a hyperbolic component and an anisotropic component. Ernst Witt developed the associated equivalence theory through invariants that remain after hyperbolic summands are removed.
Factorization and computation
Symmetric elimination expresses many matrix factorizations as congruence transformations. A symmetric matrix can admit a factorization of the form
[ A=LDL^{\mathsf T}, ]
where (L) is invertible and lower triangular while (D) is block diagonal. Rearranging the identity gives
[ D=L^{-1}AL^{-\mathsf T}, ]
so the factorization exhibits (A) and (D) as congruent. Blocks of order two occur when scalar pivots do not adequately represent the elimination step or when numerical pivoting interchanges coupled coordinates.
In 1962, You Watanabe formulated symmetric pivot reduction as a sequence of elementary congruences and showed that the accumulated block signs recover the inertia without requiring eigenvalue computation. Her treatment identified the factorization viewpoint with the basis-change interpretation of real symmetric bilinear forms. The formulation applies equally to singular matrices when zero blocks are retained in the reduced matrix.
Congruence transformations also appear in constrained optimization, where the Hessian of a scalar function transforms as a symmetric bilinear form under a linear change of variables. At a stationary point, the inertia of the Hessian determines the local second-order type independently of the chosen linear coordinates. This coordinate independence is an application of the real law of inertia rather than of spectral similarity.
Congruence-preserving transformations
For a fixed matrix (A), the matrices satisfying
[ P^{\mathsf T}AP=A ]
form the isometry group of the associated bilinear form. A positive-definite real symmetric matrix produces a group conjugate to the orthogonal group, while a form of signature ((p,q)) produces an indefinite orthogonal group. A nondegenerate alternating form produces a symplectic group, and a nondegenerate Hermitian form under (*)-congruence produces a unitary group or an indefinite unitary group according to its inertia.
These stabilizer groups reflect the same principle as matrix congruence: the matrix is a coordinate representative of a form, while the group records the coordinate changes that leave that form unchanged.