Generic Canonical Forms under *Congruence

• Generic canonical forms under *congruence are block-diagonal representations that classify most n×n complex matrices via *congruence transformations into distinct families.
• They stratify the matrix space into exactly ⌊n/2⌋+1 families based on the number of 2×2 and 1×1 blocks, ensuring robustness under perturbations.
• Their miniversal deformation structure supports analytic perturbation theory by providing local normal forms for precise control of spectral transitions.

Generic canonical forms under $^*$congruence are block-diagonal representations to which “most” $n \times n$ complex matrices can be reduced via $^*$congruence transformations. The $^*$congruence relation—$A \sim^* B$ if $B = S^* A S$ for invertible $S$—arises naturally in the classification of sesquilinear forms and governs the canonical block structure associated with matrix orbits under such transformations. Recent developments establish that, in contrast to ordinary congruence, the set of generic $^*$congruence canonical forms comprises exactly $\lfloor n/2 \rfloor + 1$ distinct families, stratified by block count and type, each open and dense within the space of $n \times n$ complex matrices (Terán et al., 13 Dec 2025). These generic forms admit a rigorous local normal form ("miniversal deformation") structure, supporting analytic perturbation theory and the stable parametrization of associated orbits (Dmytryshyn et al., 2011).

1. $^*$Congruence: Definition and Canonical Blocks

$^*$Congruence is defined for $A,B \in \mathbb{C}^{n \times n}$ by $A \sim^* B$ if there exists invertible $S$ such that $B = S^* A S$. The canonical classification by Horn and Sergeichuk asserts that every complex matrix is $^*$congruent to a direct sum of blocks in exactly three families (Dmytryshyn et al., 2011):

• Type I (Unit circle blocks): $[\mu_1, \ldots, \mu_{n-2\ell}]$, with $$\mu_j \in S^1 = \{e^{i\theta} : \theta \in [0,2\pi)\}$$, $\mu_j^2 \neq \mu_k^2$ if $j \neq k$.
• Type II (Paired blocks): $$H_2(\lambda_i) = \begin{pmatrix} 0 & 1 \ \lambda_i & 0 \end{pmatrix}$$, with $|\lambda_i|>1$, all distinct.
• Type III (Nilpotent Jordan blocks): $J_k(0)$, with canonical size determined by the underlying matrix.

Every complex matrix is $^*$congruent to a direct sum

$$A \sim^* \bigoplus_i H_{2m_i}(\lambda_i) \oplus \bigoplus_j \mu_j \Delta_{n_j} \oplus \bigoplus_\ell J_{k_\ell}(0),$$

with $|\lambda_i|>1$, $|\mu_j|=1$, and prescribed block sizes.

2. Generic $^*$Congruence Canonical Forms: Stratification

Recent work reveals a strong dichotomy between congruence and $^*$congruence in terms of generic canonical structures (Terán et al., 13 Dec 2025). For any $n \geq 1$, define $p = \lfloor n/2 \rfloor$ and parameter $\ell = 0, \ldots, p$:

• The $\ell$th generic form comprises $\ell$ Type II blocks ($2 \times 2$), and $n-2\ell$ Type I ($1 \times 1$) blocks.
• The block-diagonal canonical model for stratum $\ell$ is

$$D_\ell = H_2(\mu_1) \oplus \cdots \oplus H_2(\mu_\ell) \oplus \alpha_1 \oplus \cdots \oplus \alpha_{n-2\ell},$$

with each $\mu_i \in \mathbb{C}$, $|\mu_i|>1$, $\mu_i \neq \mu_j$ for $i \neq j$, $\alpha_j \in S^1$, and $\alpha_j^2 \neq \alpha_k^2$ for $j \neq k$.

The table below summarizes generic canonical forms for $n \times n$ matrices under $^*$congruence:

$\ell$	$2\times 2$ blocks	$1\times 1$ blocks (unit circle)
$0$	$0$	$n$
$1$	$1$	$n-2$
$2$	$2$	$n-4$
$\cdots$	$\cdots$	$\cdots$
$p$	$\lfloor n/2 \rfloor$	$n-2p$

For even $n$, $p=n/2$, $\ell$ runs $0$ to $n/2$, and for odd $n$, $\ell$ runs $0$ to $(n-1)/2$, with exactly $n-2\ell$ $1\times 1$ blocks. Each $\ell$ specifies a distinct open-dense stratum, and the union of the closures covers all $\mathbb{C}^{n\times n}$; no matrix is generic outside these configurations (Terán et al., 13 Dec 2025).

3. Structure of Parameters and Domains

In each stratum $\mathcal{G}_\ell$, the parameters fulfill strict conditions:

• Type II blocks: Each $H_2(\mu_i)$ employs a complex $\mu_i$ with modulus $|\mu_i|>1$, all distinct.
• Type I blocks: Each scalar $\alpha_j$ satisfies $\alpha_j \in S^1$, $\alpha_j^2\neq\alpha_k^2$ for $j\neq k$.

The generic canonical form thus is uniquely determined up to the values of these continuous parameters, which serve as moduli for the family. This explicit description is valid on an open subset and remains stable under small perturbations—the set of matrices with given generic canonical form $\mathcal{G}_\ell$ is open in both the Euclidean and Zariski topologies (Terán et al., 13 Dec 2025).

4. Local Normal Forms and Miniversal Deformations

Arnold’s theory, extended to $^*$congruence by Dmytryshyn, Futorny, and Sergeichuk, provides a miniversal deformation construction (Dmytryshyn et al., 2011). For any $A_{\mathrm{can}}$ in canonical form, the allowable perturbation space $\mathcal{D}(\mathbb{C})$ consists of real-linear combinations of matrices with zero entries in fixed positions and free parameters elsewhere ("stars, circles, bullets"). The minimality and analytic dependence of these parameters are encoded by the codimension of the $^*$congruence orbit: $$\dim_\mathbb{R} \mathcal{D}(\mathbb{C}) = \operatorname{codim}_\mathbb{R}\{S^*A_{\mathrm{can}}S \mid S \in GL_n(\mathbb{C})\}.$$ Blockwise, perturbations are given as

$$B_{\mathrm{mindef}} = \bigoplus_i \left(H_{2m_i}(\lambda_i)+D^H_i(x,y)\right)\oplus\cdots,$$

where each $D^H_i$ is a small real-linear combination of the independent parameters $x$, $y$ placed in the prescribed template positions.

The analytic reducing transformation $\mathcal{S}(X)$ exists and is polynomial-analytic in the entries of $X$ and $\overline{X}$, converges in operator norm, and produces a block-diagonal normal form that varies continuously with the parameters—yielding a stable smooth family for all matrices near $A_{\mathrm{can}}$ (Dmytryshyn et al., 2011).

5. Closure Relations and Perturbation Theory

The $^*$congruence canonical reduction is, in general, discontinuous; small perturbations of $A$ may jump between canonical types (1311.1144). The structure of these transitions is encoded in the closure graph: each $\mathcal{G}_\ell$ is open but its closure is disjoint from the closures of other $\mathcal{G}_{\ell'}$ for $\ell \neq \ell'$; matrices "move" between generic strata only at the boundaries (where degeneracies or block type changes occur).

Explicit analysis in the $2\times 2$ case (Futorny et al., 2013) gives the full Hasse diagram of canonical forms, each associated with a real codimension, parameter domains, and possible transitions under infinitesimal perturbation. For example, blocks of the form $O(\sigma)$, with $|\sigma|<1$, become $H(\tau)$ as $|\sigma| \to 1$, and further to scalar blocks as degeneracies occur. In higher dimensions, the generic normal forms remain robust under perturbation except at transitions where eigenvalues coalesce or leave the unit circle.

6. Significance and Implications of Generic Forms

The existence of exactly $\lfloor n/2 \rfloor + 1$ generic $^*$congruence canonical forms reflects a striking dichotomy relative to the single generic form under ordinary congruence (Terán et al., 13 Dec 2025). Each stratum is maximal in the sense that its canonical structure is locally stable—matrices therein cannot be perturbed to non-generic block patterns without losing openness. This stratification is deeply connected to the geometry of matrix spaces and the classification of sesquilinear forms; it further guarantees that most $n \times n$ matrices encountered in generic applications admit explicit reduction to one of the block-diagonal forms with continuously varying parameters.

A plausible implication is that the blockwise modularity and smooth deformation structures significantly facilitate further analysis in applications involving local inverse problems, spectral discontinuities, and model reduction in sesquilinear settings. The stratification additionally acts as a organizing principle for orbit closure relations, facilitating the study of matrix families near degeneracy loci.

7. Relation to Bilinear and Sesquilinear Forms

The correspondence between $^*$congruence canonical forms and matrix representations of sesquilinear forms in $\mathbb{C}^n \times \mathbb{C}^n$ is explicit: generic matrix representatives under $^*$congruence provide standardized templates for most forms encountered on these spaces (Terán et al., 13 Dec 2025). Each stratum $\mathcal{G}_\ell$ may be viewed as parametrizing a moduli space of forms with fixed block structure, continuous moduli given by block parameters, and clear boundaries separating degenerate cases.

This suggests that the generic reduction theory not only offers effective classification of matrix types but also encodes geometric and algebraic features relevant in more advanced studies of Hermitian geometry, invariant theory, and orbit stratification for group actions on matrix spaces.