Simultaneous diagonalization via congruence: Nondegenerate families of symmetric matrices

Technical Summary: Simultaneous Diagonalization via Congruence for Nondegenerate Families of Symmetric Matrices

Problem Statement
The paper addresses the problem of simultaneous diagonalization via congruence for a general family of real symmetric matrices (or equivalently, symmetric bilinear forms). Given a real vector space $E$ of dimension $n$ and a family of symmetric bilinear forms $\mathcal{F} = \{\gamma_t \mid t \in I\}$, the objective is to determine the necessary and sufficient conditions under which there exists a single basis $B$ of $E$ that diagonalizes every form in the family simultaneously. In matrix terms, given a family of symmetric matrices $\{A_1, \dots, A_m\}$, the goal is to find an invertible matrix $Q$ such that $Q^T A_i Q$ is diagonal for all $i$. While the case of two symmetric forms is well-studied, the paper notes that results for a general number of forms have been sparse, with significant recent contributions only emerging in the last decade.

Methodology
The author adopts a coordinate-free approach, treating symmetric matrices as representations of symmetric bilinear forms. The methodology relies on the following structural assumptions and constructions:

1. Non-degeneracy: The family $\mathcal{F}$ is assumed to be non-degenerate, meaning it contains at least one non-degenerate element $\gamma_{t_0}$. This allows for the definition of an isomorphism $\gamma_{t_0}^\flat: E \to E^*$ and its inverse $\gamma_{t_0}^\#$.
2. Linear Operators: For any $\gamma_t \in \mathcal{F}$, a linear operator $G_{t_0}(t)$ is defined on $E$ via the composition $G_{t_0}(t) := \gamma_{t_0}^\# \circ \gamma_t^\flat$. In matrix representation relative to a basis, this corresponds to $G_{t_0}(t) = [\gamma_{t_0}]^{-1} [\gamma_t]$.
3. Auxiliary Bilinear Forms: A new bilinear form $b_{t_0}(s, t)$ is introduced, defined by the relation $b_{t_0}(s, t)(u, v) := \gamma_s(u, G_{t_0}(t) \cdot v)$. The symmetry of this form is central to the analysis.

The proof strategy involves decomposing the vector space $E$ into eigenspaces of the operator $G_{t_0}(t)$. The author demonstrates that if specific algebraic conditions are met, these eigenspaces are mutually orthogonal with respect to all forms in the family, allowing for an inductive construction of a simultaneous orthogonal basis.

Key Contributions and Results
The paper establishes the following main theorem (Theorem 2) and its matrix-theoretic counterpart (Theorem 1):

Theorem 1 (Matrix Form): Let $A_1, \dots, A_m$ be real $n \times n$ symmetric matrices, and let $A_0$ be an invertible matrix formed as a linear combination of them. The family $\{A_1, \dots, A_m\}$ is simultaneously diagonalizable via congruence if and only if:

• (SD1) Each matrix $G_i := A_0^{-1} A_i$ is diagonalizable (i.e., its Jordan canonical form is diagonal).
• (SD2) For every pair $i, j$, the matrix product $b_{ij} := A_i A_0^{-1} A_j$ is symmetric (i.e., $b_{ij} = b_{ji}$).

Theorem 2 (Coordinate-Free Form): For a family of symmetric bilinear forms $\mathcal{F}$ with a non-degenerate element $\gamma_{t_0}$, the family is simultaneously diagonalizable if and only if:

• (SD1) For each $t \in I$, the operator $G_{t_0}(t)$ is diagonalizable.
• (SD2) For each pair $s, t \in I$, the bilinear form $b_{t_0}(s, t)$ is symmetric.

Corollaries and Special Cases:

• Positive/Negative Definite Case (Corollary 1 & 2): If the reference form $\gamma_{t_0}$ (or matrix $A_0$) is positive or negative definite, condition (SD1) is automatically satisfied because $G_{t_0}(t)$ becomes self-adjoint with respect to the inner product defined by $\gamma_{t_0}$. In this specific case, simultaneous diagonalizability depends only on condition (SD2) (the symmetry of the product $A_i A_0^{-1} A_j$). This result is presented as a coordinate-free equivalent to Theorem 3.3 in a previous work [12].

Significance and Claims
The paper claims to generalize existing results from the last decade regarding the simultaneous diagonalization of a general number of symmetric forms. By providing necessary and sufficient conditions that apply to non-degenerate families (not restricted to positive definite pencils), the work offers a broader algebraic framework for this problem.

The author frames the significance of this work through its potential application in cosmology, specifically noting that the research was motivated by investigating the simultaneous diagonalizability of Bianchi models. The paper positions itself as a theoretical advancement that clarifies the algebraic relations required to ensure simultaneous diagonalization, moving beyond the historically limited scope of two-form systems. The results are presented as a rigorous extension of known theorems, offering a unified condition for families of arbitrary size.