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Commuting matrices

In linear algebra, two square matrices A and B over a field (typically the real or complex numbers) are said to commute if AB = BA, meaning the order of multiplication does not affect the result.^[1] This property is expressed using the commutator [A, B] = AB - BA = 0.^[2] While matrix multiplication is generally non-commutative—unlike scalar multiplication—the commuting case is central to many advanced results in the field.^[1] A key theorem states that if A and B are both diagonalizable and commute, then there exists a single invertible matrix S such that S^{-1}AS and S^{-1}BS are both diagonal, allowing them to share a common eigenbasis.^[2] This simultaneous diagonalization simplifies computations and reveals structural similarities between the matrices.^[3] For self-adjoint (Hermitian) matrices, which arise frequently in applications like quantum mechanics and optimization, commutativity implies simultaneous diagonalization via a unitary transformation, preserving inner products and orthogonality.^[4] In quantum theory, commuting Hermitian operators represent compatible observables that can be measured simultaneously with definite outcomes, as they share common eigenvectors.^[5] More broadly, over the complex numbers, a family of commuting matrices is simultaneously upper triangularizable, with further implications for representations of algebras, Jordan canonical forms, and stability analysis in dynamical systems.^[6]

Fundamentals

Definition

In linear algebra, matrix multiplication is generally non-commutative, meaning that for arbitrary square matrices A and B of the same order n \times n, the product AB does not necessarily equal BA.^[7] Two such matrices A, B \in M_n(\mathbb{F}), where \mathbb{F} is a field (such as the real or complex numbers), are said to commute if AB = BA.^[7] This condition is equivalently expressed through the commutator [A, B] = AB - BA = 0.^[8] The concept applies specifically to square matrices of the same size, since matrix multiplication requires compatible dimensions for both AB and BA to be defined and comparable; non-square matrices do not admit a standard commutativity relation in this sense.^[7]

Basic Properties

The set of all matrices that commute with a fixed matrix A \in M_n(\mathbb{F}), denoted the centralizer C(A) = \{ B \in M_n(\mathbb{F}) \mid AB = BA \}, forms a subalgebra of the full matrix algebra M_n(\mathbb{F}) over the field \mathbb{F}. This subalgebra is closed under addition and scalar multiplication, as if B, C \in C(A) and \alpha, \beta \in \mathbb{F}, then A(\alpha B + \beta C) = \alpha AB + \beta AC = \alpha BA + \beta CA = (\alpha B + \beta C)A. It is also closed under matrix multiplication, since if B, C \in C(A), then A(BC) = (AB)C = B(AC) = B(CA) = (BC)A.^[9] The identity matrix I commutes with every matrix A, as IA = AI = A, so I \in C(A) for all A. Similarly, every scalar multiple of the identity, cI for c \in \mathbb{F}, commutes with A because (cI)A = cA = A(cI). Consequently, the centralizer of the identity is the entire matrix algebra: C(I) = M_n(\mathbb{F}).^[9] A trivial but illustrative case arises with diagonal matrices: any two diagonal matrices D_1, D_2 \in D_n(\mathbb{F}) (the algebra of n \times n diagonal matrices over \mathbb{F}) commute, as their product is entrywise multiplication of diagonals, which is commutative. In fact, the centralizer of a diagonal matrix with distinct diagonal entries is precisely the set of diagonal matrices.^[9] While commuting matrices preserve each other's eigenspaces—for if Av = \lambda v, then A(Bv) = B(Av) = \lambda (Bv), so Bv is also an eigenvector with eigenvalue \lambda unless Bv = 0—they do not necessarily share a complete set of common eigenvectors. For instance, the identity matrix commutes with every matrix but has the full space as its eigenspace, whereas a non-diagonalizable matrix like a Jordan block has a proper eigenspace of dimension less than n.^[10]

Advanced Properties

Commuting Families

A commuting family of matrices is a finite or infinite collection \{A_1, A_2, \dots, A_k\} of matrices in M_n(\mathbb{F}), where \mathbb{F} is a field, such that A_i A_j = A_j A_i for all i, j. These families extend the notion of pairwise commutativity to multiple matrices and play a key role in understanding the structure of commutative substructures within the full matrix algebra M_n(\mathbb{F}). The centralizer of a single matrix A \in M_n(\mathbb{F}), denoted Z(A), is the subalgebra consisting of all matrices B \in M_n(\mathbb{F}) that commute with A, i.e., Z(A) = \{B \in M_n(\mathbb{F}) \mid AB = BA\}. This set forms an associative algebra under matrix addition and multiplication, and any commuting family containing A is contained in Z(A). For n \times n matrices over an algebraically closed field, the dimension of Z(A) satisfies \dim Z(A) \geq n, with equality if and only if A is a cyclic matrix (i.e., the minimal polynomial of A has degree n). In this minimal case, Z(A) coincides with the algebra of all polynomials in A.^[11] A commuting family \{A_1, \dots, A_k\} generates a commutative subalgebra of M_n(\mathbb{F}), meaning the \mathbb{F}-span of all products of the A_i (including the identity) is abelian under multiplication. Such subalgebras are of particular interest because their structure reflects the joint spectral properties of the family. For instance, if the matrices are diagonalizable over \mathbb{C}, the family admits simultaneous diagonalization. A canonical example of a commuting family is the set of all diagonal n \times n matrices over \mathbb{F}, which pairwise commute since diagonal matrices commute elementwise. This family generates a commutative subalgebra of dimension n.

Simultaneous Diagonalization and Triangularization

A commuting family of diagonalizable matrices over the complex numbers can be simultaneously diagonalized via a single invertible similarity transformation. Specifically, if \{A_\alpha\} \subset M_n(\mathbb{C}) is a family of pairwise commuting diagonalizable matrices, there exists an invertible matrix S \in M_n(\mathbb{C}) such that S^{-1} A_\alpha S is diagonal for every \alpha. This result follows from the fact that such families share a common eigenbasis, allowing a unified diagonal form. For the special case of normal matrices, which are unitarily diagonalizable individually, the theorem strengthens: a family of commuting normal matrices is simultaneously unitarily diagonalizable, meaning there exists a unitary U \in M_n(\mathbb{C}) such that U^* A_\alpha U is diagonal for all \alpha. This is a direct extension of Schur's theorem to families and underscores the role of normality in preserving unitarity.^[12] More generally, any family of pairwise commuting complex matrices admits simultaneous upper triangularization, regardless of diagonalizability. That is, for commuting A, B \in M_n(\mathbb{C}), there exists an invertible P \in M_n(\mathbb{C}) such that both P^{-1} A P and P^{-1} B P are upper triangular. This holds for arbitrary finite or infinite commuting families \{A_\alpha\} \subset M_n(\mathbb{C}), where a single invertible P (or equivalently, a unitary U via equivalence of similarity and unitary triangularization) renders all P^{-1} A_\alpha P upper triangular. The diagonal entries in this form correspond to the eigenvalues, ordered compatibly across the family, reflecting shared spectral properties such as common eigenspaces. A deeper algebraic condition governs simultaneous diagonalizability: a commuting family is simultaneously diagonalizable if and only if the associative algebra it generates is semisimple. In this context, the generated algebra is commutative due to pairwise commutativity, and semisimplicity ensures decomposition into a direct sum of simple components (fields over \mathbb{C}), enabling a basis of common eigenvectors. However, not all commuting families satisfy this; for instance, a family consisting of two commuting nilpotent Jordan blocks of size greater than 1 generates a non-semisimple algebra and cannot be simultaneously diagonalized, though it remains simultaneously triangularizable. Such limitations highlight the distinction between triangular and diagonal forms in non-diagonalizable cases.

Characterizations

Algebraic Characterizations

Algebraic characterizations of commuting matrices focus on conditions expressed through polynomial identities, algebraic structures, and trace properties within the ring of matrices. A fundamental result states that if two n \times n matrices A and B over an algebraically closed field commute (i.e., AB = BA), and A is non-derogatory—meaning its minimal polynomial has degree n, equal to its characteristic polynomial—then B can be expressed as a polynomial in A of degree at most n-1. This characterization highlights how commutativity restricts the centralizer of A to the polynomial algebra generated by A itself when A achieves the maximal possible degree for its minimal polynomial. The converse also holds: if B = p(A) for some polynomial p, then AB = BA trivially, as polynomials in A commute with A. This equivalence is central to understanding the structure of commutative subalgebras in matrix rings. A related perspective arises from the algebra generated by A and B. If A and B commute, the subalgebra they generate consists of all linear combinations and products, which forms a commutative associative algebra. In particular, Weyl's work on the structure of such algebras implies that commuting matrices satisfy polynomial relations within this generated algebra, ensuring that elements derived from A and B remain interdependent through non-commutative extensions only if the original pair does not commute. This algebraic interdependence underscores that commutativity is equivalent to the generated algebra being commutative, without higher-degree relations beyond polynomials in the generators under the non-derogatory assumption. From a Lie algebra viewpoint, the commutator bracket [\cdot, \cdot] defines the Lie algebra \mathfrak{gl}(n) of n \times n matrices. If A and B commute, the Lie subalgebra generated by A and B—spanned by linear combinations xA + yB for scalars x, y—is abelian, as [xA + yB, x'A + y'B] = (xy' - yx')[A, B] = 0 for all scalars. This abelianity is a direct algebraic consequence of [A, B] = 0 and characterizes pairs where the generated Lie structure has vanishing brackets, distinguishing them from non-abelian subalgebras. Seminal treatments emphasize this as a foundational property linking matrix commutativity to broader Lie theory. An additional algebraic condition involves traces: if A and B commute, then \operatorname{tr}(A^k B^m) = \operatorname{tr}(B^m A^k) for all non-negative integers k, m, due to the cyclic property of the trace, which allows reordering A^k B^m = B^m A^k within the trace. This equality holds as a necessary consequence of commutativity but is not sufficient, as there exist non-commuting pairs satisfying it for specific k, m or even all powers in low dimensions, though counterexamples abound in general. This trace-based characterization provides a verifiable algebraic test, albeit incomplete, for potential commutativity. Finally, a variant of Frobenius's theorem offers another algebraic lens: if A and B commute and share no common invariant hyperplane (under certain irreducibility conditions on the pair), then the entire space is irreducible for the algebra they generate, implying shared invariant subspaces only when reducibility occurs. More precisely, commutativity ensures that any invariant subspace for A is also invariant for B, algebraically captured by the joint action preserving the same lattice of subspaces. This result, rooted in early invariant theory, algebraically ties commutativity to the coincidence of invariant subspace structures without invoking spectral data.^[12]

Spectral and Geometric Characterizations

Commuting matrices exhibit profound connections to spectral theory, particularly through their interactions with eigenvalues and eigenspaces. A fundamental property arises when considering an eigenvector of one matrix under the action of a commuting partner. Suppose A and B are matrices satisfying AB = BA, and let v be an eigenvector of A with eigenvalue \lambda, so Av = \lambda v. Then,

A(Bv) = B(Av) = B(\lambda v) = \lambda (Bv),

which implies that Bv is also an eigenvector of A corresponding to the same eigenvalue \lambda (or zero if Bv = 0).^[3] This demonstrates that B maps the eigenspace of A for \lambda into itself, preserving the eigenspace. More generally, if A and B commute, each preserves the eigenspaces of the other. That is, the eigenspace E_\lambda(A) = \{ v \mid Av = \lambda v \} is invariant under B, meaning B(E_\lambda(A)) \subseteq E_\lambda(A).^[3] If A and B are both diagonalizable, then they are simultaneously diagonalizable, admitting a common eigenbasis in which both are diagonal.^[3] From a geometric perspective, commutativity of linear operators on a vector space implies the existence of joint invariant subspaces. The eigenspaces of one operator serve as invariant subspaces for the other, allowing the operators to act compatibly on these subspaces. This joint invariance facilitates the decomposition of the space into common spectral components, underscoring commutativity as a condition for aligned geometric structures in the operator algebra.^[3] For non-diagonalizable matrices, the spectral characterization extends to generalized eigenspaces. If AB = BA, then B preserves each generalized eigenspace of A, defined as G_\lambda(A) = \{ v \mid (A - \lambda I)^k v = 0 \text{ for some } k \}. Within these spaces, the Jordan structures are compatible: the Jordan chains of A are mapped by B in a manner that respects the chain lengths and eigenvalue associations, enabling simultaneous upper triangularization with aligned Jordan blocks.^[13] This compatibility ensures that the nilpotent parts of the Jordan decompositions interact consistently under commutation.^[13]

Examples and Applications

Canonical Examples

One canonical class of commuting matrices consists of all diagonal matrices over the complex numbers. For any two n \times n diagonal matrices A = \diag(a_1, \dots, a_n) and B = \diag(b_1, \dots, b_n), the product is given by

AB = \diag(a_1 b_1, \dots, a_n b_n) = BA,

demonstrating commutativity directly from the absence of off-diagonal terms in the multiplication.^[3] This property holds because diagonal matrices preserve the standard basis vectors as eigenvectors, allowing simultaneous diagonalization in the same basis. In contrast, rotation matrices in three dimensions provide a fundamental example of non-commutativity. Consider the rotation matrices for 90-degree rotations around the x- and y-axes:

R_x(90^\circ) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{pmatrix}, \quad R_y(90^\circ) = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ -1 & 0 & 0 \end{pmatrix}.

Computing the products yields

R_x(90^\circ) R_y(90^\circ) = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix} \neq \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & -1 \\ -1 & 0 & 0 \end{pmatrix} = R_y(90^\circ) R_x(90^\circ).

This non-commutativity arises because the rotation group SO(3) is non-abelian, with the order of rotations affecting the final orientation.^[14] The Pauli matrices from quantum mechanics illustrate non-commutativity in a 2x2 context. Defined as

\sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad \sigma_y = \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}, \quad \sigma_z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix},

they satisfy the commutation relations [\sigma_i, \sigma_j] = 2i \epsilon_{ijk} \sigma_k, where \epsilon_{ijk} is the Levi-Civita symbol. Explicitly,

\sigma_x \sigma_y = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} = i \sigma_z = -\sigma_y \sigma_x,

so [\sigma_x, \sigma_y] = 2i \sigma_z \neq 0. However, subsets such as \{\sigma_x, \sigma_x\} or \{I, \sigma_z\} (where I is the identity) do commute.^[15] Companion matrices offer an example of a specific commuting family. For a monic polynomial p(\lambda) = \lambda^n + c_{n-1} \lambda^{n-1} + \dots + c_0, the companion matrix C is the n \times n matrix with 1's on the superdiagonal, -c_0, \dots, -c_{n-1} in the last row, and zeros elsewhere. Any matrix that commutes with C is precisely a polynomial in C, i.e., q(C) for some polynomial q. Thus, the set \{I, C, C^2, \dots, C^{n-1}\} forms a commuting family, as powers of C satisfy C^k C^m = C^{k+m} = C^m C^k.^[16] For nilpotent matrices, commuting examples require specific Jordan structures. Consider two 2x2 nilpotent Jordan blocks for eigenvalue 0, each J = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}. In a 4x4 block-diagonal form, N_1 = J \oplus J and N_2 = J \oplus J commute, as N_1 N_2 = (J^2 \oplus J^2) = N_2 N_1. However, if one is a 3x3 block J_3 = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} embedded in a larger space with a 1x1 zero block, and the other is two 2x2 blocks, they generally do not commute unless the partitions satisfy compatibility conditions (e.g., the conjugate partitions dominate each other). Such pairs are characterized by their Jordan types allowing simultaneous upper-triangularization.^[17]

Applications in Physics and Engineering

In quantum mechanics, commuting observables can be simultaneously measured because they share a common set of eigenstates.^[18] For instance, the position and momentum operators do not commute, leading to the Heisenberg uncertainty principle that prohibits precise simultaneous measurements of both.^[19] In contrast, the total angular momentum operator L^2 and its z-component L_z commute, allowing states to be labeled by simultaneous eigenvalues of both, which simplifies the description of rotational symmetries in atomic and molecular systems.^[20] Representation theory provides a framework for understanding symmetries in physical systems, where irreducible representations of abelian groups yield one-dimensional matrices that inherently commute.^[21] This property is crucial in quantum mechanics for analyzing systems invariant under abelian symmetry groups, such as translations in free particles, enabling the construction of commuting operator algebras that preserve physical observables under group actions.^[22] In control theory, commuting matrices in linear time-invariant systems facilitate stability analysis by allowing simultaneous diagonalization, which transforms the system into decoupled modes for easier eigenvalue-based assessment of asymptotic behavior.^[23] For switched linear systems, commutation relations between subsystem matrices provide sufficient conditions for overall stability, reducing the complexity of verifying Lyapunov functions across multiple modes.^[24] Signal processing leverages the commutativity of convolution operations, represented by circulant matrices, to enable efficient parallel computation of filtered signals in linear time-invariant systems.^[25] This property ensures that the order of applying multiple filters does not affect the output, optimizing algorithms for tasks like audio processing and image enhancement.^[26] Recent advancements in quantum computing exploit groups of commuting Pauli operators for gate design and Hamiltonian simulation, partitioning non-commuting terms into clusters that can be simultaneously diagonalized to minimize circuit depth and error rates.^[27] For example, post-2020 methods group Pauli strings into commuting families to accelerate variational quantum algorithms and hardware-efficient compilation on noisy intermediate-scale quantum devices.^[28] As of 2025, further advances include leveraging commuting groups within Hamiltonians for efficient variational quantum eigensolver (VQE) circuits, reducing measurement overhead.^[29]

Historical Development

Early Foundations

The discovery of non-commutative multiplication in Hamilton's quaternions, introduced by William Rowan Hamilton in 1843, marked an early challenge to the commutative laws prevalent in classical algebra and motivated subsequent explorations into non-commutative structures like matrices. Quaternions extend the complex numbers to represent rotations in three-dimensional space, forming a division algebra where multiplication is associative but not commutative, such as ij = k while ji = -k. This non-commutativity highlighted the limitations of commutative fields and spurred interest in algebraic systems capable of modeling such behaviors, laying groundwork for matrix theory as a framework for linear transformations.^[30] In 1858, Arthur Cayley formalized the theory of matrices in his seminal paper "A Memoir on the Theory of Matrices," establishing matrices as independent algebraic objects with addition, multiplication, and powers, where multiplication is generally non-commutative. A key result stated by Cayley was the Cayley-Hamilton theorem, which asserts that every square matrix A satisfies its own characteristic equation \det(\lambda I - A) = 0, implying p(A) = 0 for the characteristic polynomial p(\lambda); the first proof was provided by Frobenius in 1878. This theorem links matrices to polynomials, demonstrating that A commutes with any polynomial in itself, such as powers A^k, and provided a precursor to studying commutativity within matrix algebras.^[31]^[32] James Joseph Sylvester, who coined the term "matrix" in 1850, advanced matrix theory in the 1880s through lectures and papers at Johns Hopkins University, introducing concepts like the adjugate and nullity that underpin the structure of matrix commutants. His work on the "centralizer" of a matrix—the set of all matrices commuting with a given one—emerged in this period, framing questions about the dimensionality and form of commutative subalgebras within the full matrix ring. This built on Cayley's foundations, emphasizing how non-commutative matrix multiplication contrasts with commutative scalar algebras while identifying subspaces where commutativity holds.^[32] Ferdinand Frobenius contributed foundational results on commuting matrices in the late 1870s and 1890s, culminating in his 1878 theorem that any finite set of commuting complex matrices can be simultaneously upper triangularized via a similarity transformation. This result, proved using induction on the dimension and properties of irreducible representations, revealed that commuting families share compatible eigenspaces, distinguishing them from general non-commuting sets. Frobenius's work, alongside earlier efforts by Cayley and Sylvester, filled a critical pre-20th-century gap by shifting focus from isolated matrices to families, underscoring the interplay between commutativity and simultaneous canonical forms in non-commutative algebras.^[12]

Modern Extensions

In 1909, Issai Schur formalized aspects of the simultaneous triangularization of commuting families of matrices over the complex numbers, establishing that any such family can be simultaneously upper triangularized via a similarity transformation, extending his earlier work on the Schur decomposition for single matrices. This result relies on the existence of a common invariant subspace and the application of Schur's lemma, which asserts that operators commuting with an irreducible representation are scalar multiples of the identity. For commuting normal matrices, simultaneous diagonalization via a unitary transformation is possible.^[12] Schur's framework provided a foundational tool for analyzing the structure of commuting operators in finite-dimensional spaces. During the 1920s, Hermann Weyl integrated these ideas into the representation theory of compact Lie groups, showing that irreducible unitary representations lead to commuting families of matrices that can be simultaneously diagonalized, leveraging the complete reducibility of representations for compact groups. Weyl's approach, detailed in works such as his 1927 paper on spectra of finite groups and subsequent developments in his 1931 book on classical groups, emphasized the role of commuting matrices in decomposing representations into simultaneous eigenspaces, bridging linear algebra with group theory. In the 2010s, numerical methods for computing centralizers—the sets of matrices commuting with a given matrix—advanced through extensions of LAPACK routines solving the associated Sylvester equation AX - XA = 0. These include the TRSYL routine for triangular cases and iterative solvers like the Bartels-Stewart algorithm, which reduce the problem via Schur decomposition and compute the null space efficiently, with LAPACK version 3.7 (2016) incorporating improved stability for large-scale computations. Such algorithms enable practical applications in control theory and optimization, achieving high accuracy for matrices up to order 1000 on modern hardware.^[33] Commuting matrices play a key role in the structure of abelian subgroups of the general linear group GL(n, ℂ), where maximal abelian subalgebras correspond to simultaneously triangularizable families, aiding the classification of semisimple Lie algebras and their representations.^[34] Post-2000 developments highlight connections to quantum information theory, particularly in stabilizer codes for quantum error correction, where the stabilizer group is generated by a commuting set of Pauli matrices, allowing fault-tolerant quantum computation by encoding logical qubits in joint eigenspaces.^[35] For instance, the [[7,1,3]] Steane code uses six commuting Pauli operators to correct single-qubit errors, demonstrating the practical impact in fault-tolerant quantum systems.

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