Generic Canonical Form Under Congruence

• Generic canonical form under congruence is defined as an algebraic object in a dense open subset where its structure is determined by a minimal set of continuously parametrized invariants.
• It synthesizes classical block decompositions with miniversal deformations to enable smooth, local reduction and classification of matrices under congruence.
• The theory bridges linear and multilinear algebra by connecting perturbation analysis, stability, and bifurcation phenomena to canonical block structures.

A matrix or algebraic object is said to be in generic canonical form under congruence if it belongs to a dense open subset—typically in the standard or Zariski topology—such that the congruence class is determined by a minimal set of continuously parametrized invariants. In matrix theory, congruence refers to the equivalence relation $A \sim B$ if $B = S^\mathrm{T} A S$ for some invertible $S$. The generic canonical form characterizes the underlying structure of “most” matrices (in the sense of measure or topology) under this action, providing a powerful synthesis of canonical-type theorems and deformation theory in linear and multilinear algebra.

1. Definition and Principal Theorems

For $A\in\mathbb{C}^{n\times n}$, two matrices are congruent if $B = S^{\mathrm{T}} A S$ for $S\in\mathrm{GL}_n(\mathbb{C})$. The Horn–Sergeichuk congruence classification asserts that every $A$ is congruent to a block-direct sum of indecomposable canonical blocks: skew-symmetric Jordan blocks $H_m(\lambda)$, symmetric nilpotent blocks $K_m$, and long Kronecker blocks $L_m$ (1311.1144). The generic canonical form under congruence is realized as the unique canonical form representing an open dense subset of $\mathbb{C}^{n\times n}$, with canonical blocks and parameters fully described in (Terán et al., 13 Dec 2025).

The existence of a miniversal deformation of a given matrix $A$ asserts that there is a holomorphic family of congruence transformations that smoothly and locally reduce any perturbation of $A$ to a canonical form plus a minimal number of independent parameters (“stars”). The number of such parameters equals the codimension of the congruence orbit, i.e., $n^2-\dim T(A)$, where $$T(A) = \{X^\mathrm{T}A + AX : X\in\mathbb{C}^{n\times n}\}$$ (1311.1144).

2. Explicit Description of the Generic Canonical Form

Complex Case

For $A\in\mathbb{C}^{n\times n}$, let $n=2m$ (even) or $n=2m+1$ (odd). The set

• $$G = \left\{A: A \sim_{congr} \bigoplus_{i=1}^m H_2(\mu_i),\ |\mu_i|>1\text{ or }\mu_i = e^{i\theta_i},\ 0<\theta_i<\pi,\,\mu_i \neq \mu_j\right\}$$ (is augmented by a $[1]$ block when $n$ is odd) is open and dense in the Frobenius-topology, and its closure is all of $\mathbb{C}^{n\times n}$ (Terán et al., 13 Dec 2025). Thus, the unique generic congruence canonical form for complex $n\times n$ matrices is
• $n=2m$:

$$A \sim P \left[\bigoplus_{i=1}^{m} H_2(\mu_i)\right] P^\mathrm{T}$$

• $n=2m+1$:

$$A \sim P \left[\bigoplus_{i=1}^{m} H_2(\mu_i) \oplus [1]\right] P^\mathrm{T}$$

where

$$H_2(\mu) = \begin{pmatrix} 0 & 1 \ \mu & 0 \end{pmatrix}$$

and the $\mu_i$ are distinct and satisfy $|\mu_i|>1$ or $\mu_i=e^{i\theta_i}$, $0<\theta_i<\pi$.

This form is unique (up to permutations of the $\mu_i$) and corresponds to the codimension-zero congruence orbit (Terán et al., 13 Dec 2025).

Real and Other Fields

The real case involves subtlety: the generic canonical structure for real congruence comprises only $2\times2$ and $4\times4$ blocks associated with real and nonreal eigenvalues, respectively, subject to spectral and symmetry constraints (Terán et al., 8 Oct 2025). The structure over fields of characteristic $\neq2$ for other algebraic objects (e.g., pairs of symmetric and skew-symmetric matrices) follows analogous reduction schemes (Bovdi et al., 2017, Bovdi et al., 2017).

3. Miniversal Deformations and Stability

The canonical form under congruence is discontinuous with respect to matrix entries, rendering the orbit structure highly stratified. Despite this, the miniversal deformation (or ‘analytic normal form’) provides a continuous, holomorphic parametrization of all matrices in a neighborhood of any $A$, via

$S(X)^\mathrm{T}(A+X)S(X) = A_\text{can} + D(X)$

where $A_\text{can}$ is the block-canonical form, $S(0)=I$, and $D(X)$ contains exactly codim$\,T(A)$ analytic parameters (1311.1144).

For generic $A$, each $H_2(\mu)$ block contributes one free complex parameter, corresponding to the local coordinate transversal to the congruence orbit. This establishes codimension as the precise count of moduli required to parametrize generic perturbations, and gives a local model for orbit structure, stratification, and bifurcations.

4. Connection with Multilinear and Polynomial Canonical Forms

Canonical forms under congruence generalize to multilinear contexts, such as families of symmetric matrices (Fang et al., 3 Mar 2025), pairs of skew-symmetric matrices (Bovdi et al., 2017), or forms of higher degree (Reznick, 2012). The guiding principle is that the generic congruence class is characterized by a minimal block structure and parameter set, captured by rank, eigenvalue, or other spectral data.

For example, the generic canonical form for a simultaneous block diagonalization of symmetric matrices is governed by the structure of the joint “center” algebra and the theory of orthogonal idempotents (Fang et al., 3 Mar 2025).

In the polynomial case, a general (Zariski-open) quartic in two variables is congruent to the sum of a quadratic square and a fourth power of a linear form, and genericity is verified by parameter-count and Jacobian rank arguments (Reznick, 2012).

5. Perturbations, Algebraic Bundles, and Instability

The unique open dense congruence class is robust under generic perturbation, but the boundaries of this generality are precisely where canonical form jumps occur (codimension increases, or degeneracies such as coincident eigenvalues). Miniversal deformations organize such bifurcations, capturing the possible transitions between different canonical types and indicating the “moduli” space of congruence classes in the vicinity of a given matrix (1311.1144, Dmytryshyn, 2011).

In the case of pairs of skew-symmetric matrices, miniversal deformations elucidate how minimal indices and eigenvalue data can coalesce or split under perturbation, further structuring the local congruence geometry (Dmytryshyn, 2011).

6. Standard-Form Congruence and Nonclassical Equivalences

Beyond the classical bilinear congruence, generalizations such as standard-form congruence—arising from affine linear changes in the context of nonhomogeneous quadratics—yield their own generic normal forms, with new classification types reflecting nonreducible linear or constant terms. The classification for $3\times3$ matrices under standard-form congruence organizes algebras defined by two generators and a single quadratic relation, with a finite number of distinct generic types (Gaddis, 2012).

7. Generic Canonical Forms in Broader Settings

Extensions to real matrices, Hermitian or $*$-congruence, and multilinear structures adapt the block-type correspondences and parameter regimes to account for field-specific properties and invariant types (Terán et al., 8 Oct 2025, Dmytryshyn et al., 2011, Fang et al., 3 Mar 2025). In all these cases, the philosophy of “generic” means open-dense (in standard or analytic topology), minimal block structure, and parameters subject only to general nonsingularity or noncoincidence constraints.

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The theory of generic canonical form under congruence—across matrices, symmetric or skew-symmetric pairs, higher degree forms, and related algebraic structures—constitutes the unifying normal form underpinning deformation, perturbation theory, moduli classification, and the local geometry of orbit spaces in linear algebra (Terán et al., 13 Dec 2025, 1311.1144, Bovdi et al., 2017, Reznick, 2012, Fang et al., 3 Mar 2025).