Self-inverse linear subspaces of matrices

Abstract: We study linear subspaces of matrices whose inverse spaces are also linear. Based on the fact that any linear space containing the identity matrix and whose inverse space is linear must be a self-inverse space, we introduce such spaces as self-inverse spaces. In fact, as we will show, self-inverse spaces are finite-dimensional complex unitary Jordan algebras. We provide an algebraic classification of all self-inverse spaces of small type and a classification of small-dimensional self-inverse spaces up to isomorphism.

• The paper shows that self-inverse subspaces are exactly Jordan algebras containing the identity, providing a rigorous structural characterization.
• Key methodologies include Peirce decomposition and an analysis of minimal polynomial multiplicities to classify subspaces.
• Results offer explicit classifications for dimensions 1–4 and insights on representation theory with significant implications for algebraic statistics.

Self-Inverse Linear Subspaces of Matrices

Scope and Motivation

This paper provides a structural and algebraic analysis of linear subspaces $L \subset \mathbb{C}^{n\times n}$ whose inverse spaces $L^{-1}$ are also linear. By leveraging the fact that any linear space containing the identity and possessing a linear inverse space must be self-inverse, i.e., $L = L^{-1}$, the authors formalize and entirely characterize such subspaces as finite-dimensional unitary Jordan algebras. The work addresses foundational questions at the intersection of linear algebra, Jordan algebra theory, and algebraic statistics, particularly in contexts where covariance and concentration matrices in Gaussian graphical models are constrained to the same linear space.

Structural Characterization

The foundational lemmas establish that for a linear space $L$ containing $\operatorname{Id}$, if $L^{-1}$ is linear, then $L$ is closed under all non-negative matrix powers and is self-inverse. Crucially, self-inverse spaces coincide with Jordan subalgebras of $\mathbb{C}^{n\times n}$ containing the identity, equipped with the usual Jordan product $X*Y = \frac{1}{2}(XY + YX)$.

Further, the equivalence of several characterizations is proven:

• $L$ is self-inverse if and only if $L^{-1}$0 for all $L^{-1}$1 and $L^{-1}$2;
• closure under the Jordan product $L^{-1}$3 for all $L^{-1}$4 is equivalent to self-inverseness.

The authors analyze structural decomposition via Peirce decomposition, showing that a Jordan algebra admits a unique decomposition into irreducible subalgebras, with the uniqueness theorem proved for products of irreducible Jordan algebras.

Minimal Polynomial Types and Algebraic Classification

A core result is the isomorphism classification of self-inverse spaces generated by single matrices: $L^{-1}$5 if and only if their minimal polynomials have the same number of roots with identical multiplicities. This yields a classification of self-inverse spaces of small type, defining canonical varieties $L^{-1}$6 corresponding to minimal polynomial root multiplicities.

The paper elaborates on the algebraic geometry of these spaces by describing associated varieties $L^{-1}$7, which encode the loci of matrices with prescribed root multiplicities. The intersection theory and dimensional invariants reveal how combinatorial and algebraic constraints translate into global subspace classification.

Explicit Classification up to Dimension Four

An exhaustive classification is presented for self-inverse spaces up to dimension four (see Theorems on dimensions 1–4), listing canonical representatives and identifying their isomorphism types. Notably:

• One-dimensional: only the identity.
• Two-dimensional: $L^{-1}$8 and $L^{-1}$9.
• Three-dimensional: six non-isomorphic spaces, including upper-triangular and symmetric $L = L^{-1}$0 matrices.
• Four-dimensional: sixteen canonical spaces, including entirely explicit forms for spaces like $L = L^{-1}$1, $L = L^{-1}$2, $L = L^{-1}$3, and various block structures.

Strong claims are made regarding the finiteness and structure of these classes, supported by explicit algebraic calculations.

Representation Theory and Orbit Structure

The representation theory of SIS (self-inverse spaces) is developed, examining the action of $L = L^{-1}$4 and conjugacy classes. Every abstract SIS admits representations in symmetric matrices, showing the relevance to classical settings. There exist SIS with infinitely many irreducible representations and, for $L = L^{-1}$5, infinitely many non-conjugate orbits. Ideals and quotient constructions are analyzed, elucidating the possible kernel structures and furthering the isomorphism classification within Jordan algebras.

Implications and Open Problems

The implications for algebraic statistics are significant: models where covariance and concentration matrices lie in the same space enable rational ML estimators and well-behaved ML degrees. The classification provides building blocks for such models, with the correspondence between internal matrix symmetries (via minimal polynomials) and global statistical constraints fully elucidated.

Theoretically, the results connect linear algebra with Jordan algebra theory, clarifying the structure and uniqueness of Jordan algebra decompositions and offering a complete resolution for small dimension cases. The identification of algebraic varieties controlling minimal polynomial types suggests further directions in algebraic geometry and representation theory.

Open questions include:

• Classification in higher dimensions.
• Finiteness and enumeration of $L = L^{-1}$6-dimensional self-inverse spaces.
• Asymptotics or bounds in the count of such spaces.
• Orbit structure under matrix groups and its relation to the geometry of algebraic varieties.

Conclusion

This work provides a rigorous algebraic and geometric framework for understanding self-inverse linear spaces of matrices, situating them as unitary Jordan algebras and delivering a comprehensive classification up to dimension four. The analysis blends structure theory, representation theory, and algebraic geometry, establishing foundational results with practical implications for algebraic statistics and future research into matrix subspace classification and Jordan algebra representation theory.