Linear Commuting & Diagonalizable Updates

• Linear commuting and diagonalizable updates are linear transformations where commuting matrices are simultaneously diagonalizable, enabling unified spectral decomposition.
• They simplify complex dynamics through simultaneous linearization, streamline solutions of matrix equations, and bolster control system analysis.
• They underpin practical algorithms in quantum information and numerical stability, linking algebraic structure with computational performance.

Linear commuting and diagonalizable updates refer to the interplay between algebraic structure (commutativity), spectral structure (diagonalizability), and the properties of linear transformations or update maps in finite and infinite-dimensional settings. This topic is central to the paper of local dynamics, stability analysis, invariant theory, control, quantum information, and the computational solution of matrix equations. The following sections provide a comprehensive overview across foundational theory, formal criteria, algorithmic approaches, matrix equations, practical implications, and recent developments.

1. Foundational Structure: Commuting and Diagonalizable Operators

Linear maps or matrices that commute and are diagonalizable exhibit profound regularity. If a family of complex (or real) matrices $\{A_1, \ldots, A_h\}$ commutes pairwise and each is diagonalizable, then they are simultaneously diagonalizable—there exists a single invertible matrix $S$ such that $S^{-1} A_j S$ is diagonal for each $j$ (1005.3434, 1007.1983, 2308.05450). The same applies to general linear transformations of finite-dimensional vector spaces and, with some caveats, to certain infinite-dimensional settings.

Simultaneous diagonalizability underpins the concept of linear commuting updates: if each $A_j$ represents a repeated linear action (or update), their joint action can be analyzed in a common spectral basis. The algebra generated by a set of commuting, diagonalizable matrices is thus itself diagonalizable, and every element is a polynomial in the generating matrices with coefficients determined by the spectral data (1501.05190, 2506.08695).

Commuting normal matrices (Hermitian, unitary, or more generally normal) are always simultaneously diagonalizable by a unitary transformation. For general diagonalizable matrices, simultaneous diagonalizability requires only commutativity (1005.3434, 1007.1983, 2006.16364, 2308.05450).

2. Simultaneous Linearization and Joint Diagonalization Criteria

Formal Linearization in Complex Dynamics

Given $h$ formal germs $f_1,\ldots, f_h$ of biholomorphisms fixing the origin in $\mathbb{C}^n$ with respective linear parts $A_1,\ldots, A_h$, a formal (or holomorphic) simultaneous linearization is a power series $y(z) = z + \{\text{higher order terms}\}$ such that $f_k \circ y = y \circ A_k$ for all $k$ (1005.3434). Linearizability is determined by:

• Algebraic structure: If the $A_j$ are almost simultaneously Jordanizable and commute, formal simultaneous linearization exists.
• Diagonalizability: If $A_1,\ldots,A_h$ are simultaneously diagonalizable, commutativity alone suffices.
• Small divisor conditions: For convergence (holomorphic linearization), a Brjuno-type arithmetic condition on the spectra of $A_1,\ldots, A_h$ ensures the power series converges.

The above gives a multidimensional answer to classical problems (e.g., Moser's question) in local holomorphic dynamics (1005.3434).

Approximate and Algorithmic Joint Diagonalization

When matrices nearly commute ($\|[A,B]\|$ small), they are nearly jointly diagonalizable. For self-adjoint (Hermitian) matrices, the degree of almost commutativity controls how nearly a unitary change of basis can diagonalize them both, quantified by joint off-diagonal "energy" measures (1305.2135, 2205.15519). Constructive algorithms (Jacobi-type, vector-wise, Riemannian Newton methods) have been developed to find approximate joint diagonalizers, with convergence guarantees connected to commutator norms and classical theorems such as Lin's theorem.

For families of matrices, block methods construct a common eigenbasis by sequentially diagonalizing blocks associated with repeated eigenvalues (2006.16364). In cases where exact commutativity does not hold, techniques yield small perturbations to commuting, exactly diagonalizable counterparts whose error is controlled directly by the norm of their commutators (2205.15519).

3. Algebraic Characterizations, Invariants, and Preserver Results

Matrix Invariants and Representation Theory

The algebra of invariants for $m$-tuples of commuting $n \times n$ matrices under conjugation is isomorphic, via restriction, to the algebra of $\mathbb{S}_n$-invariants on $m$ diagonal matrices (1501.05190). That is, all invariants of commuting matrices are determined by their spectra up to permutation. This reflects the deep connection between commutativity (simultaneous diagonalizability) and the symmetry inherent in the action of the symmetric group on diagonal entries.

Linear Preservers and Subspace Structure

Automorphisms of the full matrix algebra $M_n(\mathbb{R})$ that preserve the set of diagonalizable matrices must be compositions of inner automorphisms (conjugation), transposition, and additive scalar-valued maps (1007.1983). The maximal linear subspaces of diagonalizable matrices are conjugate to the symmetric matrices, and the structure of simultaneously diagonalizable pairs is captured by the intersection of such subspaces, leading to further insights in control, spectral algorithms, and optimization.

Conjugation-Invariant Subsets and Nilpotent Structures

Subsets of matrices invariant under conjugation and closed under linear combinations of commuting elements are described in terms of their spectral and Jordan block structure (2205.06147). For diagonalizable matrices over algebraically closed fields, only a finite list of such subsets arises, and for nilpotents, membership is governed by Jordan cell size patterns, enabling a precise criterion for inclusion based on combinatorial conditions.

4. Matrix Equations and Linear Updates with Commuting Parameters

Consistency and Solution Structure

When the parameters of a linear matrix equation (e.g., Sylvester, Stein, Lyapunov equations) form a commuting set of diagonalizable matrices, solution theory simplifies significantly (2406.09429). The following equivalence is established: the equation is consistent if and only if a certain matrix, constructed from the Drazin (or group) inverse and the spectra of the commuting operators, is a solution; equivalently, the standard associated linear matrix equation is consistent. The dimension of the affine space of solutions is determined by the zero entries in a matrix built from the eigenvalue data ("relevant matrix"). This framework gives explicit formulas for solution spaces and connects naturally to the underlying simultaneous diagonalization.

The set of all possible diagonalizing matrices for a commuting sequence is described in terms of block structures, induced vector sequences, and permutations, which is crucial for constructing solution sets in explicit computational algorithms.

5. Practical Algorithms, Perturbations, and Quantum Computation

Simultaneous Diagonalization in Computation

Efficient numeric methods for simultaneous diagonalization exploit the structure of commuting, diagonalizable matrices to avoid expensive matrix inversions or large Kronecker product systems (2110.11133). Newton-type and Riemannian optimization algorithms deliver quadratic convergence under suitable spectral separation and well-conditioned initialization.

Stability under Perturbations

Joint eigenvalue perturbation bounds generalizing the Hoffman–Wielandt theorem are established for tuples of commuting normal or diagonalizable matrices (1710.05215). The relative change in joint spectrum is controlled by the Frobenius norm of relative perturbations and by condition numbers of the diagonalizations. Clifford algebra methods facilitate the extension of these results to tuples of matrices.

Quantum algorithms for computing eigenvalues of diagonalizable matrices deploy quantum phase estimation, ODE simulation, and singular value estimation, yielding quantum states that encode spectral decompositions. For $s$-sparse matrices, these algorithms have complexity polynomial in $s$, eigenvalue spread, problem conditioning, and inverse precision, with performance dictated by the conditioning of the eigenvector basis and the spectral properties of the updates (1912.08015).

Block Diagonalization for Non-Commuting Sets

In cases where unitary matrices are not fully commuting (as in physical or representation-theoretic contexts), simultaneous block diagonalization is the best possible outcome. Algorithms exist that, given known decompositions into common invariant subspaces, construct explicit transfer matrices effecting the block diagonalization, with applications in particle physics for understanding outer automorphism actions (2012.14440).

6. Applications in Control, Lie Algebras, and Quantum Information

Ensemble Controllability

Uniform ensemble controllability for linear systems with diagonalizable drift is characterized via module-theoretic lifting of the reachable set and the rank condition of the ensemble controllability matrix (2112.14301). This algebraic framework generalizes previously known results and applies to both real (RR) and complex conjugate pairs (RC) ensembles by checking the controllability of finite-dimensional induced subsystems determined by spectral properties.

Commuting Maps and Biderivations in Algebras

Commuting linear maps and skew-symmetric biderivations on Lie algebras, especially in the context of deformative Schrödinger–Virasoro algebras and more general centerless or perfect Lie algebras, are classified in terms of the centroid: under weak assumptions, all commuting linear maps are centroidal, and non-inner biderivations give rise to non-standard commuting maps, enriching symmetry and update structure (1611.05718, 1801.01109).

Quantum Processes and Non-Demolition Measurements

If a family of Kraus operators (describing quantum channel updates) are commuting and satisfy standard normalization, each is normal and the entire family is simultaneously diagonalizable (2308.05450). This has a direct impact on the analysis of quantum measurements, enabling diagonal reduction of dynamics, simplifying the description of non-demolition measurement processes, and ensuring the existence of underlying observables that generate the commuting update family.

7. Difference Schemes, Frobenius Commuting, and Finite Field Enumeration

Matrices over finite fields commuting with their Frobenius automorphism (the map $M \mapsto \sigma(M)$ raising entries to the $p$-th power) are studied with respect to both their diagonalizability and their commutation with all elements of their Frobenius orbit (2506.08695). For diagonalizable matrices, the count of those commuting with all their Frobenius transforms is governed by the number of $\mathbb{F}_p$-split maximal commutative subalgebras, leading to asymptotic formulas of the form

$$|{}_∞^{(\text{diag})} \cap M_n(\mathbb{F}_q)| = p^{n^2 - n} \cdot q^n + O_{p,n}(q^{n-1}),$$

while the broader (not necessarily diagonalizable) set is counted by higher exponents determined via combinatorial and algebraic geometry methods. This analysis links difference algebra, matrix commutation varieties, and arithmetic geometry, including applications to the paper of wild ramification in number theory.

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This synthesis demonstrates the pervasive influence of commuting and diagonalizable updates in linear algebra, dynamics, control theory, invariant theory, number theory, and quantum information science. The underlying principles translate into efficient algorithms, structural decompositions, and spectral stability analyses, all facilitated by the core properties of commutativity and diagonalizability.