On the Necessary and Sufficient Condition for a Set of Matrices to Commute and Some Further Linked Results

Abstract

This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [𝐴,𝐵]=0 for two matrices 𝐴 and 𝐵 if and only if a vector 𝑣(𝐵) defined uniquely from the matrix 𝐵 is in the null space of a well-structured matrix defined as the Kronecker sum 𝐴⊕(−𝐴^∗), which is always rank defective. This result is extendable directly to any countable set of commuting matrices. Complementary results are derived concerning the commutators of certain matrices with functions of matrices 𝑓(𝐴) which extend the well-known sufficiency-type commuting result [𝐴,𝑓(𝐴)]=0.

1. Introduction

The problem of commuting operators and matrices, in particular, is very relevant in a significant number of problems of several branches of science, which are very often mutually linked, cited herein after.

(1) In several fields of interest in Applied Mathematics or Linear Algebra [1–22] including Fourier transform theory, graph theory where, for instance, the commutativity of the adjacency matrices is relevant [1, 17–19, 21–35], Lyapunov stability theory with conditional and unconditional stability of switched dynamic systems involving discrete systems, delayed systems, and hybrid systems where there is a wide class of topics covered including their corresponding adaptive versions including estimation schemes (see, e.g., [23–41]). Generally speaking, linear operators, and in particular matrices, which commute share some common eigenspaces. On the other hand, a known mathematical result is that two graphs with the same vertex set commute if their adjacency matrices commute [16]. Graphs are abstract representations of sets of objects (vertices) where some pairs of them are connected by links (arcs/edges). Graphs are often used to describe behaviors of multiconfiguration switched systems where nodes represent each parameterized dynamics and arcs describe allowed switching transitions [35]. They are also used to describe automatons in Computer Science. Also, it has been proven that equalities of products involving two linear combinations of two any length products having orthogonal projectors (i.e., Hermitian idempotent matrices) as factors are equivalent to a commutation property [21].

(2) In some fields in Engineering, such as multimodel regulation and Parallel multiestimation [36–41]. Generally speaking switching among configurations can improve the transient behavior. Switching can be performed arbitrarily (i.e., at any time instant) through time while guaranteeing closed-loop stability if a subset of the set of configurations is stable provided that a common Lyapunov function exists for them. This property is directly related to certain pair wise commutators of matrices describing configuration dynamics being zero [7, 10, 11, 14, 15]. Thus, the problem of commuting matrices is in fact of relevant interest in dynamic switched systems, namely, those which possess several parameterized configurations, one of them, is becoming active at each current time interval. If the matrices of dynamics of all the parameterizations commute then there exists a common Lyapunov function for all those parameterizations and any arbitrary switching rule operating at any time instant maintains the global stability of the switched rule provided that all the parameterizations are stable [7]. This property has been described also in [23–25, 28–30] and many other references therein. In particular, there are recent studies which prove that, in these circumstances, arbitrary switching is possible if the matrices of dynamics of the various configurations commute while guaranteeing closed-loop stability. This principle holds not only in both the continuous-time delay-free case and in the discrete-time one but even in configurations involving time-delay and hybrid systems as well. See, for instance, [10–15, 27–30, 34–41] and references therein. The set of involved problems is wide enough like, for instance, switched multimodel techniques [27–30, 35, 36, 40, 41], switched multiestimation techniques with incorporated parallel multiestimation schemes involving adaptive control [34, 38–40], time delay and hybrid systems with several configurations under mutual switching, and so forth [10, 11, 14, 15] and references therein. Multimodel tools and their adaptive versions incorporating parallel multiestimation are useful to improve the regulation and tracking transients including those related to triggering circuits with regulated transient via multiestimation [36], master-slave tandems [39], and so forth. However, it often happens that there is no common Lyapunov function for all the parameterizations becoming active at certain time intervals. Then, a minimum residence (or dwelling) time at each active parameterization has to be respected before performing the next switching in order to guarantee the global stability of the whole switched system so that the switching rule among distinct parameterizations is not arbitrary [7, 12, 13, 27–30, 34–41].

(3) In some problems of Signal Processing. See, for instance, [1, 17, 18] concerning the construction of DFT (Discrete Fourier transform)-commuting matrices. In particular, a complete orthogonal set of eigenvectors can be obtained for several types of offset DFT’s and DCT’s under commutation properties.

(4) In certain areas of Physics, and in particular, in problems related to Quantum Mechanics. See, for instance, [22, 42, 43]. Basically, a complete set of commuting observables is a set of commuting operators whose eigenvalues completely specify the state of a system since they share eigenvectors and can be simultaneously measured [22, 42, 43]. These Quantum Mechanics tools have also inspired other Science branches. For instance, it is investigated in the above mentioned reference [18] a commuting matrix whose eigenvalue spectrum is very close to that of the Gauss-Hermite differential operator. It is proven that it furnishes two generators of the group of matrices which commute with the discrete Fourier transform. It is also pointed out that the associate research inspired in Quantum Mechanics principles. There is also other relevant basic scientific applications of commuting operators. For instance, the symmetry operators in the point group of a molecule always commute with its Hamiltonian operator [20]. The problem of commuting matrices is also relevant to analyze the normal modes in dynamic systems or the discussion of commuting matrices dependent on a parameter (see, e.g., [2, 3]).

It is well known that commuting matrices have at least a common eigenvector and also, a common generalized eigenspace [4, 5]. A less restrictive problem of interest in the above context is that of almost commuting matrices, roughly speaking, the norm of the commutator is sufficiently small [5, 6]. A very relevant related result is that the sum of matrices which commute is an infinitesimal generator of a 𝐶₀-semigroup. This leads to a well-known result in Systems Theory establishing that the matrix function 𝑒^{𝐴1𝑡1+𝐴2𝑡2}=𝑒^𝐴1𝑡1𝑒^𝐴2𝑡2 is a fundamental (or state transition) matrix for the cascade of the time invariant differential systems 𝑥̇₁(𝑡)=𝐴₁𝑥₁(𝑡), operating on a time 𝑡₁, and 𝑥̇₂(𝑡)=𝐴₂𝑥₂(𝑡), operating on a time 𝑡₂, provided that 𝐴₁ and 𝐴₂ commute (see, e.g., [7–11]).

Most of the abundant existing researches concerning sets of commuting operators, in general, and matrices, in particular, are based on the assumption of the existence of such sets implying that each pair of mutual commutators is zero. There is a gap in giving complete conditions guaranteeing that such commutators within the target set are zero. This paper formulates the necessary and sufficient condition for any countable set of (real or complex) matrices to commute. The sequence of obtained results is as follows. Firstly, the commutation of two real matrices is investigated in Section 2. The necessary and sufficient condition for two matrices to commute is that a vector defined uniquely from the entries of any of the two given matrices belongs to the null space of the Kronecker sum of the other matrix and its minus transpose. The above result allows a simple algebraic characterization and computation of the set of commuting matrices with a given one. It also exhibits counterparts for the necessary and sufficient condition for two matrices not to commute. The results are then extended to the necessary and sufficient condition for commutation of any set of real matrices in Section 3. In Section 4, the previous results are directly extended to the case of complex matrices in two very simple ways, namely, either by decomposing the associated algebraic system of complex matrices into two real ones or by manipulating it directly as a complex algebraic system of equations. Basically, the results for the real case are directly extendable by replacing transposes by conjugate transposes. Finally, further results concerning the commutators of matrices with matrix functions are also discussed in Section 4. The proofs of the main results in Sections 2, 3, and 4 are given in corresponding Appendices A, B, and C. It may be pointed out that there is implicit following duality of the main result. Since a necessary and sufficient condition for a set of matrices to commute is formulated and proven, the necessary and sufficient condition for a set of matrices not to commute is just the failure in the above one to hold.

1.1. Notation

[𝐴,𝐵] is the commutator of the square matrices 𝐴 and 𝐵.

𝐴⊗𝐵∶=(𝑎_𝑖𝑗𝐵) is the Kronecker (or direct) product of 𝐴∶=(𝑎_𝑖𝑗) and 𝐵.

𝐴⊕𝐵∶=𝐴⊗𝐼_𝑛+𝐼_𝑛⊗𝐵 is the Kronecker sum of the square matrices 𝐴∶=(𝑎_𝑖𝑗) and both of order 𝑛, where 𝐼_𝑛 is the nth identity matrix.

𝐴^𝑇 is the transpose of the matrix 𝐴 and 𝐴^∗ is the conjugate transpose of the complex matrix 𝐴. For any matrix 𝐴, Im𝐴 and Ker𝐴 are its associate range (or image) subspace and null space, respectively. Also, rank(𝐴) is the rank of 𝐴 which is the dimension of Im(𝐴) and det(𝐴) is the determinant of the square matrix 𝐴.

𝑣(𝐴)=(𝑎^𝑇1,𝑎^𝑇2,…,𝑎^𝑇𝑛)^𝑇∈𝐂^𝑛2 if 𝑎^𝑇𝑖∶=(𝑎_𝑖1,𝑎_𝑖2,…,𝑎_𝑖𝑛) is the ith row of the square matrix 𝐴.

𝜎(𝐴) is the spectrum of 𝐴;𝑛∶={1,2,…,𝑛}. If 𝜆_𝑖∈𝜎(𝐴) then there exist positive integers 𝜇_𝑖 and 𝜈_𝑖≤𝜇_𝑖 which are, respectively, its algebraic and geometric multiplicity; that is, the times it is repeated in the characteristic polynomial of 𝐴 and the number of its associate Jordan blocks, respectively. The integer 𝜇≤𝑛 is the number of distinct eigenvalues and the integer 𝑚_𝑖, subject to 1≤𝑚_𝑖≤𝜇_𝑖, is the index of 𝜆_𝑖∈𝜎(𝐴); ∀𝑖∈𝜇, that is, its multiplicity in the minimal polynomial of 𝐴.

𝐴∼𝐵 denotes a similarity transformation from 𝐴 to 𝐵=𝑇⁻¹𝐴𝑇 for given 𝐴,𝐵∈𝐑^𝑛×𝑛 for some nonsingular 𝑇∈𝐑^𝑛×𝑛. 𝐴≈𝐵=𝐸𝐴𝐹 means that there is an equivalence transformation for given 𝐴,𝐵∈𝐑^𝑛×𝑛 for some nonsingular 𝐸,𝐹∈𝐑^𝑛×𝑛.

A linear transformation from 𝐑^𝑛 to 𝐑^𝑛, represented by the matrix 𝑇∈𝐑^𝑛×𝑛, is denoted identically to such a matrix in order to simplify the notation. If 𝑉≠Dom𝑇≡𝐑^𝑛 is a subspace of 𝐑^𝑛 then Im𝑇(𝑉)∶={𝑇𝑧∶𝑧∈𝑉} and Ker𝑇(𝑉)∶={𝑧∈𝑉∶𝑇𝑧=0∈𝐑^𝑛}. If 𝑉≡𝐑^𝑛, the notation is simplified to Im𝑇∶={𝑇𝑧∶𝑧∈𝐑^𝑛} and Ker𝑇∶={𝑧∈𝐑^𝑛∶𝑇𝑧=0∈𝐑^𝑛}.

The symbols “⋀’’ and “∨’’ stand for logic conjunction and disjunction, respectively. The abbreviation “iff’’ stands for “if and only if.’’ The notation card 𝑈 stands for the cardinal of the set 𝑈. 𝐶_𝐴 (resp., 𝐶_𝐴) is the set of matrices which commute (resp., do not commute) with a matrix 𝐴. 𝐶_𝐀 (resp., 𝐶_𝐀) is the set of matrices which commute (resp., do not commute) with all square matrix 𝐴_𝑖 belonging to a given set 𝐀.

2. Results Concerning the Sets of Commuting and No Commuting Matrices with a Given One

Consider the sets 𝐶_𝐴∶={𝑋∈𝐑^𝑛×𝑛∶[𝐴,𝑋]=0}≠∅, of matrices which commute with A, and 𝐶_𝐴∶={𝑋∈𝐑^𝑛×𝑛∶[𝐴,𝑋]≠0}, of matrices which do not commute with 𝐴; ∀𝐴∈𝐑^𝑛×𝑛. Note that 0∈𝐑^𝑛×𝑛∩𝐶_𝐴; that is, the zero n-matrix commutes with any n-matrix so that, equivalently, 0∉𝐑^𝑛×𝑛∩𝐶_𝐴 and then 𝐶_𝐴∩𝐶_𝐴=∅; ∀𝐴∈𝐑^𝑛×𝑛. The subsequent two basic results which follow are concerned with commutation and noncommutation of two real matrices 𝐴 and 𝑋. The used tool relies on the calculation of the null space and the range space of the Kronecker sum of the matrix 𝐴, one of the matrices, with its minus transpose. A vector built with all the entries of the other matrix 𝑋 has to belong to one of the above spaces for 𝐴 and 𝑋 to commute and to the other one in order that 𝐴 and 𝑋 not to be two commuting matrices.

Proposition 2.1. (i) 𝐶_𝐴={𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈Ker(𝐴⊕(−𝐴^𝑇))}.
(ii) 𝐶_𝐴=𝐑^𝑛×𝑛⧵C_A={𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∉Ker(𝐴⊕(−𝐴^𝑇))}≡{𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈Im(𝐴⊕(−𝐴^𝑇))}.
(iii) 𝐵∈𝐶_𝐴∶={𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈Ker(𝐴⊕(−𝐴^𝑇))}.

Note that according to Proposition 2.1 the set of matrices 𝐶_𝐴 which commute with the square matrix 𝐴 and its complementary 𝐶_𝐴 (i.e., the set of matrices which do not commute with 𝐴) can be redefined in an equivalent way by using their given expanded vector forms.

Proposition 2.2. One has rank𝐴⊕−𝐴^𝑇<𝑛²⟺⟺0K∈er𝜎𝐴𝐴⊕⊕−−𝐴𝐴^𝑇_𝑇≠0⟺∃𝑋(≠0)∈𝐶_𝐴,∀𝐴∈𝐑^𝑛×𝑛.(2.1)

Proof. One has [𝐴,𝐴]=0;∀𝐴∈𝐑^𝑛×𝑛⇒∃𝐑^𝑛2∋0≠𝑣(𝐴)∈Ker(𝐴⊕(−𝐴^𝑇)); ∀𝐴∈𝐑^𝑛×𝑛. As a result, Ker𝐴⊕−𝐴^𝑇≠0∈𝐑^𝑛2;∀𝐴∈𝐑^𝑛×𝑛⟺rank𝐴⊕−𝐴^𝑇<𝑛²;∀𝐴∈𝐑^𝑛×𝑛(2.2) so that 0∈𝜎(𝐴⊕(−𝐴^𝑇)).
Also, ∃𝑋(≠0)∈𝐑^𝑛×𝑛∶[𝐴,𝑋]=0⇔𝑋∈𝐶_𝐴 since Ker(𝐴⊕(−𝐴^𝑇))≠0∈𝐑^𝑛2.
Then, Proposition 2.2 has been proved.

The subsequent mathematical result is stronger than Proposition 2.2 and is based on characterization of the spectrum and eigenspaces of 𝐴⊕(−𝐴^𝑇).

Theorem 2.3. The following properties hold.
(i) The spectrum of 𝐴⊕(−𝐴^𝑇) is 𝜎(𝐴⊕(−𝐴^𝑇))={𝜆_𝑖𝑗=𝜆_𝑖−𝜆_𝑗∶𝜆_𝑖,𝜆_𝑗∈𝜎(𝐴);∀𝑖,𝑗∈𝑛} and possesses 𝜈 Jordan blocks in its Jordan canonical form of, subject to the constraints 𝑛²≥𝜈=dim𝑆=(∑^𝜇𝑖=1𝜈_𝑖)²≥𝜈(0), and 0∈𝜎(𝐴⊕(−𝐴^𝑇)) with an algebraic multiplicity 𝜇(0) and with a geometric multiplicity 𝜈(0) subject to the constraints: 𝑛²=^𝜇_𝑖=1𝜇_𝑖²≥𝜇(0)≥^𝜇_𝑖=1𝜇^2𝑖≥𝜈(0)=^𝜇_𝑖=1𝜈^2𝑖≥𝑛,(2.3) where
(a)𝑆∶=span{𝑧_𝑖⊗𝑥_𝑗,∀𝑖,𝑗∈𝑛}, 𝜇_𝑖=𝜇(𝜆_𝑖) and 𝜈_𝑖=𝜈(𝜆_𝑖) are, respectively, the algebraic and the geometric multiplicities of 𝜆_𝑖∈𝜎(𝐴), ∀𝑖∈𝑛; 𝜇≤𝑛 is the number of distinct 𝜆_𝑖∈𝜎(𝐴)(𝑖∈𝜇), 𝜇_𝑖=𝜇(𝜆_𝑖𝑗) and 𝜈_𝑖𝑗=𝜈(𝜆_𝑖𝑗), are, respectively, the algebraic and the geometric multiplicity of 𝜆_𝑖𝑗=(𝜆_𝑖−𝜆_𝑗)∈𝜎(𝐴⊕(−𝐴^𝑇)), ∀𝑖,𝑗∈𝑛; 𝜇≤𝑛,(b)𝑥_𝑗 and 𝑧_𝑖 are, respectively, the right eigenvectors of 𝐴 and 𝐴^𝑇 with respective associated eigenvalues 𝜆_𝑗 and 𝜆_𝑖;∀𝑖,𝑗∈𝑛.
(ii) One has dimIm𝐴⊕−𝐴^𝑇==𝑛ra₂nk𝐴⊕−𝐴^𝑇−𝜈(0)⟺dimKer𝐴⊕−𝐴^𝑇=𝜈(0);∀𝐴∈𝐑^𝑛×𝑛.(2.4)

Expressions which calculate the sets of matrices which commute and which do not commute with a given one are obtained in the subsequent result.

Theorem 2.4. The following properties hold.
(i) One has 𝑋∈⟺𝐶_𝐴𝑋iff∈𝐶𝐴_𝐴⊕−𝐴^𝑇𝑣(𝑋)=0iff𝑣(𝑋)=−𝐹𝑣^𝑇𝑋₂𝐴^𝑇12𝐴^−𝑇11,𝑣^𝑇𝑋₂^𝑇(2.5) for any 𝑣(𝑋₂)∈Ker(𝐴₂₂−𝐴₂₁𝐴⁻¹¹¹𝐴₁₂), where 𝐸,𝐹∈𝐑^𝑛2×𝑛2 are permutation matrices and 𝑋∈𝐑^𝑛×𝑛 and 𝑣(𝑋)∈𝐑^𝑛2 are defined as follows.
(a)One has 𝑣𝑋∶=𝐹⁻¹𝑣(𝑋),𝐴⊕−𝐴^𝑇≈𝐴∶=𝐸𝐴⊕−𝐴^𝑇𝐹,∀𝑋∈𝐶_𝐴,(2.6) where 𝑣(𝑋)=(𝑣^𝑇(𝑋₁),𝑣^𝑇(𝑋₂))^𝑇∈𝐑^𝑛2 with 𝑣(𝑋₁)∈𝐑^𝜈(0) and 𝑣(𝑋₂)∈𝐑^{𝑛2−𝜈(0)}.(b)The matrix 𝐴₁₁∈𝐑^{𝜈(0)×𝜈(0)} is nonsingular in the block matrix partition 𝐴∶=Blockmatrix(𝐴_𝑖𝑗;𝑖,𝑗∈2) with 𝐴₁₂∈𝐑^{𝜈(0)×𝑛2}, 𝐴₂₁∈𝐑^{(𝑛2−𝜈(0))×𝜈(0)} and 𝐴₂₂∈𝐑^{(𝑛2−𝜈(0))×(𝑛2−𝜈(0))}.
(ii) 𝑋∈𝐶_𝐴, for any given 𝐴(≠0)∈𝐑^𝑛×𝑛, if and only if 𝐴⊕−𝐴^𝑇𝑣(𝑋)=𝑣(𝑀)(2.7) for some 𝑀(≠0)∈𝐑^𝑛×𝑛 such that rank𝐴⊕−𝐴^𝑇=rank𝐴⊕−𝐴^𝑇,𝑣(𝑀)=𝑛²−𝜈(0).(2.8) Also, 𝐶_𝐴∶=r𝑋ank∈𝐑𝐴⊕^𝑛×𝑛∶−𝐴𝐴_𝑇⊕−𝐴^𝑇𝑣(𝑋)=𝑣(𝑀)forany𝑀(≠0)∈𝐑^𝑛×𝑛satisfying=rank𝐴⊕−𝐴^𝑇,𝑣(𝑀)=𝑛²−𝜈(0).(2.9) Also, with the same definitions of 𝐸, 𝐹, and 𝑋 in (i), 𝑋∈𝐶_𝐴 if and only if 𝑣(𝑋)=𝐹𝑣^𝑇𝑀₁𝐴^−𝑇11−𝑣^𝑇𝑋₂𝐴^𝑇12𝐴^−𝑇11,𝑣^𝑇𝑋₂^𝑇,(2.10) where 𝑣(𝑋₂) is any solution of the compatible algebraic system 𝐴₂₂−𝐴₂₁𝐴⁻¹¹¹𝐴₁₂𝑣𝑋₂=𝑣𝑀₂−𝐴₂₁𝐴⁻¹¹¹𝑣𝑀₁(2.11) for some 𝑀(≠0)∈𝐑^𝑛×𝑛 such that 𝑋,𝑀∈𝐑^𝑛×𝑛 which are defined according to 𝑣(𝑋)=𝐹𝑣(𝑋) and 𝑀=𝐸𝑀𝐹≈𝑀(≠0)∈𝐑^𝑛×𝑛 with 𝑣(𝑀)=𝐸𝑣(𝑀)=𝐸(𝑣^𝑇1(𝑀),𝑣^𝑇2(𝑀))^𝑇.

3. Results Concerning Sets of Pair Wise Commuting Matrices

Consider the following sets.

(1)A set of nonzero 𝑝≥2 distinct pair wise commuting matrices 𝐀_𝐶∶={𝐴_𝑖∈𝐑^𝑛×𝑛∶[𝐴_𝑖,𝐴_𝑗]=0;∀𝑖,𝑗∈𝑝}.(2)The set of matrices MC_𝐀𝐶∶={𝑋∈𝐑^𝑛×𝑛∶[𝑋,𝐴_𝑖]=0;∀𝐴_𝑖∈𝐀_𝐶} which commute with the set 𝐀_𝐶 of pair wise commuting matrices.(3)A set of matrices 𝐶_𝐀∶={𝑋∈𝐑^𝑛×𝑛∶[𝑋,𝐴_𝑖]=0;∀𝐴_𝑖∈𝐀} which commute with a given set of nonzero 𝑝 matrices 𝐀∶={𝐴_𝑖∈𝐑^𝑛×𝑛;∀𝑖∈𝑝} which are not necessarily pair wise commuting.

The complementary sets of MC_𝐀𝐶 and 𝐶_𝐀 are MC_𝐀𝐶 and 𝐶_𝐀, respectively, so that 𝐑^𝑛×𝑛∋𝐵∈MC_𝐀𝐶 if 𝐵∉MC_𝐀𝐶 and 𝐑^𝑛×𝑛∋𝐵∈𝐶_𝐀 if 𝐵∉𝐶_𝐀. Note that 𝐶_𝐀𝐶=MC_𝐴𝐶 for a set of pair wise commuting matrices 𝐀_𝐶 so that the notation MC_𝐴𝐶 is directly referred to a set of matrices which commute with all those in a set of pair wise commuting matrices. The following two basic results are concerned with the commutation and noncommutation properties of two matrices.

Proposition 3.1. The following properties hold. (i)One has 𝐴_𝑖∈𝐀_𝐶;∀𝑖∈𝑝⟺⟺𝑣𝑣𝐴𝐴_𝑖𝑖∈_{𝑗(≠𝑖)∈𝑝}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝑖∈𝑝∈_{𝑖+1≤𝑗≤𝑝}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝑖∈𝑝.(3.1)(ii)Define 𝑁∶=_𝑖𝐀𝐴_𝐶_𝑇1⊕−𝐴₁𝐴^𝑇2⊕−𝐴₂⋯𝐴^𝑇𝑖−1⊕−𝐴_𝑖−1𝐴^𝑇𝑖+1⊕−𝐴_𝑖+1⋯𝐴^𝑇𝑝⊕−𝐴_𝑝^𝑇∈𝐑^{(𝑝−1)𝑛2×𝑛2}.(3.2) Then 𝐴_𝑖∈𝐀_𝐶;∀𝑖∈𝑝 if and only if 𝑣(𝐴_𝑖)∈Ker𝑁_𝑖(𝐀_𝐶);∀𝑖∈𝑝.(iii)One has MC_𝐀𝐶∶=⎧⎪⎨⎪⎩𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈_𝑖∈𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖;𝐴_𝑖∈𝐀_𝐶=𝑋∈𝐑_𝑛×𝑛⎫⎪⎬⎪⎭∶𝑣(𝑋)∈Ker𝑁𝐀_𝐶⊃𝐶_𝐀𝐶⊃𝐀_𝐶⊃{0}∈𝐑^𝑛×𝑛,(3.3) where 𝑁(𝐀_𝐶)∶=[𝐴^𝑇1⊕(−𝐴₁)𝐴^𝑇2⊕(−𝐴₂)⋯𝐴^𝑇𝑝⊕(−𝐴_𝑝)]^𝑇∈𝐑^{𝑝𝑛2×𝑛2},𝐴_𝑖∈𝐀_𝐶.(iv)One has MC_𝐀𝐶∶=⎧⎪⎨⎪⎩𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈_𝑖∈𝑝Im𝐴_𝑖⊕−𝐴^𝑇𝑖;𝐴_𝑖∈𝐀_𝐶=𝑋∈𝐑_𝑛×𝑛⎫⎪⎬⎪⎭∶𝑣(𝑋)∈Im𝑁𝐀_𝐶.(3.4)(v)One has 𝐶_𝐀=∶=⎧⎪⎨⎪⎩𝑋𝑋∈∈𝐑𝐑^𝑛×𝑛_𝑛×𝑛∶𝑣(𝑋)∈_𝑖∈𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖;𝐴_𝑖∈𝐀⎫⎪⎬⎪⎭∶𝑣(𝑋)∈Ker𝑁(𝐀),(3.5) where 𝑁(𝐀)∶=[𝐴^𝑇1⊕(−𝐴₁)𝐴^𝑇2⊕(−𝐴₂)⋯𝐴^𝑇𝑝⊕(−𝐴_𝑝)]^𝑇∈𝐑^{𝑝𝑛2×𝑛2},𝐴_𝑖∈𝐀.(vi)One has 𝐶_𝐀∶==⎧⎪⎨⎪⎩𝑋𝑋∈∈𝐑𝐑^𝑛×𝑛_𝑛×𝑛∶𝑣(𝑋)∈_𝑖∈𝑝Im𝐴_𝑖⊕−𝐴^𝑇𝑖;𝐴_𝑖∈𝐀⎫⎪⎬⎪⎭∶𝑣(𝑋)∈Im𝑁(𝐀).(3.6)

Concerning Proposition 3.1(v)-(vi), note that if 𝑋∈𝐶_𝐀, then 𝑋≠0 since 𝐑^𝑛×𝑛∋0∈𝐶_𝐀. The following result is related to the rank defectiveness of the matrix 𝑁(𝐀_𝐶) and any of their submatrices since 𝐀_𝐶 is a set of pair wise commuting matrices.

Proposition 3.2. The following properties hold: 𝑛²>rank𝑁𝐀_𝐶≥rank𝑁_𝑖𝐀_𝐶≥rank𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝐴_𝑗∈𝐀_𝐶;∀𝑖,𝑗∈𝑝(3.7) and, equivalently, det𝑁^𝑇𝐀_𝐶𝑁𝐀_𝐶==0de;t∀𝑁𝐴^𝑇𝑖_𝑗𝐀_𝐶𝑁_𝑖𝐀_𝐶=det𝐴_𝑗⊕−𝐴^𝑇𝑗∈𝐀_𝐶;∀𝑖,𝑗∈𝑛.(3.8)

Results related to sufficient conditions for a set of matrices to pair wise commute are abundant in literature. For instance, diagonal matrices are always pair wise commuting. Any sets of matrices obtained via multiplication by real scalars with any given arbitrary matrix are sets of pair wise commuting matrices. Any set of matrices obtained by linear combinations of one of the above sets consists also of pair wise commuting matrices. Any matrix commutes with any of its matrix functions, and so forth. In the following, a simple, although restrictive, sufficient condition for rank defectiveness of 𝑁(𝐀) of some set 𝐀 of 𝑝 square real 𝑛-matrices is discussed. Such a condition may be useful as a practical test to elucidate the existence of a nonzero 𝑛-square matrix which commutes with all matrices in this set. Another useful test obtained from the following result relies on a necessary condition to elucidate if the given set consists of pair wise commuting matrices.

Theorem 3.3. Consider any arbitrary set of nonzero 𝑛-square real matrices 𝐀∶={𝐴₁,𝐴₂,…,𝐴_𝑝} for any integer 𝑝≥1 and define matrices: 𝑁𝑁_𝑖((𝐀𝐀))∶∶==𝐴𝐴^𝑇1_𝑇1⊕−𝐴₁𝐴^𝑇2⊕−𝐴₂⋯𝐴^𝑇𝑖−1⊕−𝐴_𝑖−1𝐴^𝑇𝑖+1⊕−𝐴_𝑖+1⋯𝐴^𝑇𝑝⊕−𝐴_𝑝^𝑇,⊕−𝐴₁𝐴^𝑇2⊕−𝐴₂⋯𝐴^𝑇𝑝⊕−𝐴_𝑝^𝑇.(3.9) Then, the following properties hold: (i)rank(𝐴_𝑖⊕(−𝐴_𝑖))≤rank𝑁_𝑖(𝐀)≤rank𝑁(𝐀)<𝑛²;∀𝑖∈𝑝.(ii)⋂_𝑖∈𝑝Ker(𝐴_𝑖⊕(−𝐴^𝑇𝑖))≠{0} so that ∃𝑋(≠0)𝑋∈𝐶∈_𝐀,𝑋∈𝐶_𝐀⟺𝑣(𝑋)∈_𝑖∈𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖,𝐶_𝐀⟺𝑣(𝑋)∈_𝑖∈𝑝Im𝐴_𝑖⊕−𝐴^𝑇𝑖.(3.10)(iii)If 𝐀=𝐀_𝐶 is a set of pair wise commuting matrices then 𝑣𝐴_𝑖⟺∈𝑣𝐴_{𝑗∈𝑝⧵𝑖𝑖}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝑖∈𝑝⟺𝑣𝐴_𝑖∈_𝑖∈𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖;∀𝑖∈𝑝∈_{𝑖∈𝑝⧵{𝑖}}Ker𝐴_𝑖⊕−𝐴^𝑇𝑖;∀𝑖∈𝑝.(3.11)(iv)One has M𝐀_𝐶∶=⎧⎪⎨⎪⎩𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)_𝑖∈𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖,∀𝐴_𝑖∈𝐀_𝐶⎫⎪⎬⎪⎭⊃𝐀_𝐶∪{0}∈𝐑^𝑛×𝑛(3.12) with the above set inclusion being proper.

Note that Theorem 3.3(ii) extends Proposition 3.1(v) since it is proved that 𝐶_𝐀⧵{0}≠∅ because all nonzero 𝐑^𝑛×𝑛∋Λ=diag(𝜆𝜆⋯𝜆)∈𝐶_𝐀 for any 𝐑∋𝜆≠0 and any set of matrices 𝐀. Note that Theorem 3.3(iii) establishes that 𝑣(𝐴_𝑖)∈⋂_{𝑖∈𝑝⧵{𝑖}}Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗));∀𝑖∈𝑝 is a necessary and sufficient condition for the set to be a set of commuting matrices 𝐀 being simpler to test (by taking advantage of the symmetry property of the commutators) than the equivalent condition 𝑣(𝐴_𝑖)∈⋂_𝑖∈𝑝Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗));∀𝑖∈𝑝. Further results about pair wise commuting matrices or the existence of nonzero commuting matrices with a given set are obtained in the subsequent result based on the Kronecker sum of relevant Jordan canonical forms.

Theorem 3.4. The following properties hold for any given set of 𝑛-square real matrices 𝐀={𝐴₁,𝐴₂,…,𝐴_𝑝}.
(i) The set 𝐶_𝐀 of matrices 𝑋∈𝐑^𝑛×𝑛 which commute with all matrices in 𝐀 is defined by: 𝐶_𝐀∶=𝑋∈𝐑^𝑛×𝑛∶𝑣(𝑋)∈^𝑝_𝑖=1Ker𝐽^𝐴_𝑖=𝑋∈𝐑_𝑛×𝑛⊕−𝐽^𝑇𝐴_𝑖𝑃^𝑖−1⊗𝑃^𝑖−𝑇∶𝑣(𝑋)∈^𝑝_𝑖=1Im𝑃_𝑖=𝑋∈𝐑_𝑛×𝑛⊗𝑃^𝑖−1𝑌_𝑖∧𝑌_𝑖∈Ker𝐽_𝐴𝑖⊕−𝐽^𝑇𝐴_𝑖;∀𝑖∈𝑝∶𝑣(𝑋)∈^𝑝_𝑖=1Im𝑃_𝑖⊗𝑃^𝑖−1(𝑌),𝑌∈^𝑝_𝑖=1Ker𝐽^𝐴_𝑖⊕−𝐽^𝑇𝐴_𝑖,(3.13) where 𝑃_𝑖∈𝐑^𝑛×𝑛 is a nonsingular transformation matrix such that 𝐴_𝑖∼𝐽_𝐴𝑖=𝑃^𝑖−1𝐴_𝑖𝑃_𝑖, 𝐽_𝐴𝑖 being the Jordan canonical form of 𝐴_𝑖.
(ii) One has dimspan𝑣(𝑋)∶𝑋∈𝐶_𝐀≤=mmiin_𝑖∈𝑝n_𝑖∈𝑝dimKer𝐽_𝐴𝑖⊕−𝐽^𝑇𝐴_𝑖𝜈_𝑖(0)=min_𝑖∈𝑝^𝜌𝑖_𝑗=1𝜈^2𝑖𝑗≤min^𝑖∈𝑝^𝜌𝑖_𝑖=1𝜇^2𝑖𝑗≤min^𝑖∈𝑝𝜇_𝑖(0),(3.14) where 𝜈_𝑖(0) and 𝜈_𝑖𝑗 are, respectively, the geometric multiplicities of 0∈𝜎(𝐴_𝑖⊕(−𝐴^𝑇𝑖)) and 𝜆_𝑖𝑗∈𝜎(𝐴_𝑖) and 𝜇_𝑖(0) and 𝜇_𝑖𝑗 are, respectively, the algebraic multiplicities of 0∈𝜎(𝐴_𝑖⊕(−𝐴^𝑇𝑖)) and 𝜆_𝑖𝑗∈𝜎(𝐴_𝑖); ∀𝑗∈𝜌_𝑖 (the number of distinct eigenvalues of 𝐴_𝑖), ∀𝑖∈𝑝.
(iii) The set 𝐀 consists of pair wise commuting matrices, namely 𝐶_𝐀=MC_𝐀, if and only if 𝑣(𝐴_𝑗)∈⋂^{𝑝𝑖(≠𝑗)=1}(Ker[(𝐽_𝐴𝑖⊕(−𝐽^𝑇𝐴_𝑖))(𝑃^𝑖−1⊗𝑃^𝑖−𝑇)]); ∀𝑗∈𝑝. Equivalent conditions follow from the second and third equivalent definitions of 𝐶_𝐀 in Property (i).

Theorems 3.3 and 3.4 are concerned with MC_𝐀≠{0}∈𝐑^𝑛×𝑛 for an arbitrary set of real square matrices A and for a pair wise-commuting set, respectively.

4. Further Results and Extensions

The extensions of the results for commutation of complex matrices are direct in several ways. It is first possible to decompose the commutator in its real and imaginary part and then apply the results of Sections 2 and 3 for real matrices to both parts as follows. Let 𝐴=𝐴_re+𝐢𝐴_im and 𝐵=𝐵_re+𝐢𝐵_im be complex matrices in 𝐂^𝑛×𝑛 with 𝐴_re and 𝐵_re being their respective real parts, and 𝐴_im and 𝐵_im, all in 𝐑^𝑛×𝑛, their respective imaginary parts, and 𝐢=√−1 is the imaginary complex unity. Direct computations with the commutator of 𝐴 and 𝐵 yield [𝐴,𝐵]=𝐴_re,𝐵_re−𝐴_im,𝐵_im+𝐢𝐴_im,𝐵_re+𝐴_re,𝐵_im.(4.1) The following three results are direct and allow to reduce the problem of commutation of a pair of complex matrices to the discussion of four real commutators.

Proposition 4.1. One has 𝐵∈𝐶_𝐴⇔(([𝐴_re,𝐵_re]=[𝐴_im,𝐵_im])⋀([𝐴_im,𝐵_re]=[𝐵_im,𝐴_re])).

Proposition 4.2. One has (𝐵_re∈(𝐶_𝐴re∩𝐶_𝐴im)⋀𝐵_im∈(𝐶_𝐴im∩𝐶_𝐴re))⇒𝐵∈𝐶_𝐴.

Proposition 4.3. One has (𝐴_re∈(𝐶_𝐵re∩𝐶_𝐵im)⋀𝐴_im∈(𝐶_𝐵im∩𝐶_𝐵re))⇒𝐵∈𝐶_𝐴.

Proposition 4.1 yields to the subsequent result.

Theorem 4.4. The following properties hold.
(i) Assume that the matrices 𝐴 and 𝐵_re are given. Then, 𝐵∈𝐶_𝐴 if and only if 𝐵_im satisfies the following linear algebraic equation: 𝐴^re𝐴_im⊕−𝐴^𝑇_re⊕−𝐴^𝑇_im𝑣𝐵^re=𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re𝑣𝐵^im(4.2) for which a necessary condition is rank𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re=rank𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re𝐴^re𝐴_im⊕−𝐴^𝑇_re⊕−𝐴^𝑇_im𝑣𝐵^re.(4.3)
(ii) Assume that the matrices 𝐴 and 𝐵_im𝑒 are given. Then, 𝐵∈𝐶_𝐴 if and only if 𝐵_re satisfies (4.2) for which a necessary condition is rank𝐴^re𝐴_im⊕−𝐴^𝑇_re⊕−𝐴^𝑇_im=rank𝐴^re𝐴_im⊕−𝐴^𝑇_re⊕−𝐴^𝑇_im𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re𝑣𝐵^im.(4.4)
(iii) Also, ∃𝐵≠0 such that 𝐵∈𝐶_𝐴 with 𝐵_re=0 and ∃𝐵≠0 such that 𝐵∈𝐶_𝐴 with 𝐵_im=0.

A more general result than Theorem 4.4 is the following.

Theorem 4.5. The following properties hold.
(i) 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛 if and only if 𝑣(𝐵) is a solution to the following linear algebraic system: 𝐴^re𝐴_im⊕−𝐴^𝑇_re−𝐴_im⊕𝐴^𝑇_im⊕−𝐴^𝑇_im−𝐴_re⊕𝐴^𝑇_re𝑣𝐵^re𝑣𝐵_im=0.(4.5) Nonzero solutions 𝐵∈𝐶_𝐴, satisfying 𝑣𝐵^re𝑣𝐵_im∈Ker𝐴^re𝐴_im⊕−𝐴^𝑇_re−𝐴_im⊕𝐴^𝑇_im⊕−𝐴^𝑇_im−𝐴_re⊕𝐴^𝑇_re,(4.6) always exist since Ker𝐴^re𝐴_im⊕−𝐴^𝑇_re−𝐴_im⊕𝐴^𝑇_im⊕−𝐴^𝑇_im−𝐴_re⊕𝐴^𝑇_re≠{0}∈𝐑^2𝑛2,(4.7) and equivalently, since rank𝐴^re𝐴_im⊕−𝐴^𝑇_re−𝐴_im⊕𝐴^𝑇_im⊕−𝐴^𝑇_im−𝐴_re⊕𝐴^𝑇_re<2𝑛².(4.8)
(ii) Property (ii) is equivalent to 𝐵∈𝐶_𝐴⟺𝐴⊕−𝐴^∗𝑣(𝐵)=0(4.9) which has always nonzero solutions since rank(𝐴⊕(−𝐴^∗))<𝑛².

The various results of Section 3 for a set of distinct complex matrices to pair wise commute and for characterizing the set of complex matrices which commute with those in a given set may be discussed by more general algebraic systems like the above one with four block matrices 𝐴^𝑗re𝐴_𝑗im⊕−𝐴^𝑇2_re−𝐴_𝑗im⊕𝐴^𝑇𝑗_im⊕−𝐴^𝑇2_im−𝐴𝑗_2re⊕𝐴^𝑇𝑗_re(4.10) for each 𝑗∈𝑝 in the whole algebraic system. Theorem 4.5 extends directly for sets of complex matrices commuting with a given one and complex matrices commuting with a set of commuting complex matrices as follows.

Theorem 4.6. The following properties hold.
(i) Consider the sets of nonzero distinct complex matrices 𝐀∶={𝐴_𝑖∈𝐂^𝑛×𝑛∶𝑖∈𝑝} and 𝐶_𝐀∶={𝑋∈𝐂^𝑛×𝑛∶[𝑋,𝐴_𝑖]=0;𝐴_𝑖∈𝐀,∀𝑖∈𝑝} for 𝑝≥2. Thus, 𝐶_𝐀∋𝑋=𝑋_re+𝐢𝑋_re if and only if ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐴𝐴^1re_1im⊕−𝐴^𝑇1_re−𝐴_1im⊕𝐴^𝑇1_im𝐴_2re⊕−𝐴^𝑇1_im−𝐴_1re⊕𝐴^𝑇1_re𝐴_2im⊕−𝐴^𝑇2_re−𝐴_2im⊕𝐴^𝑇2_im𝐴_𝑝re⊕−𝐴^𝑇2_im⋮−𝐴_2re⊕𝐴^𝑇2_re𝐴_𝑝im⊕−𝐴^𝑇𝑝_re−𝐴_𝑝im⊕𝐴^𝑇𝑝_im⊕−𝐴^𝑇𝑝_im−𝐴^𝑝re⊕𝐴^𝑇𝑝_re⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦𝑣𝑋^re𝑣𝑋_im=0,(4.11) and a nonzero solution 𝑋∈𝐶_𝐀 exists since the rank of the coefficient matrix of (4.11) is less than 2𝑛².
(ii) Consider the sets of nonzero distinct commuting complex matrices 𝐀_𝐶∶={𝐴_𝑖∈𝐂^𝑛×𝑛∶𝑖∈𝑝} and MC_𝐀∶={𝑋∈𝐂^𝑛×𝑛∶[𝑋,𝐴_𝑖]=0;𝐴_𝑖∈𝐀,∀𝑖∈𝑝} for 𝑝≥2. Thus, MC_𝐀∋𝑋=𝑋_re+𝐢𝑋_re if and only if 𝑣(𝑋_r𝑒) and 𝑣(𝑋_im) are solutions to (4.11).
(iii) Properties (i) and (ii) are equivalently formulated by from the algebraic set of complex equations: 𝐴^∗1⊕−𝐴₁𝐴^∗2⊕−𝐴₂⋯𝐴^∗𝑝⊕−𝐴_𝑝^∗𝑣(𝑋)=0.(4.12)

Remark 4.7. Note that all the proved results of Sections 2 and 3 are directly extendable for complex commuting matrices, by simple replacements of transposes by conjugate transposes, without requiring a separate decomposition in real and imaginary parts as discussed in Theorems 4.5(ii) and 4.6(iii).

Let 𝑓∶𝐂→𝐂 be an analytic function in an open set 𝐷⊃𝜎(𝐴) for some matrix 𝐴∈𝐂^𝑛×𝑛 and let 𝑝(𝜆) be a polynomial fulfilling 𝑝^(𝑖)(𝜆_𝑘)=𝑓^(𝑖)(𝜆_𝑘); ∀𝑘∈𝜎(𝐴), ∀𝑖∈𝑚_𝑘−1∪{0}; ∀𝑘∈𝜇 (the number of distinct elements in 𝜎(𝐴)), where 𝑚_𝑘 is the index of 𝜆_𝑘, that is, its multiplicity in the minimal polynomial of 𝐴. Then, 𝑓(𝐴) is a function of a matrix 𝐴 if 𝑓(𝐴)=𝑝(𝐴), [8]. Some results follow concerning the commutators of functions of matrices.

Theorem 4.8. Consider a nonzero matrix 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛 for any given nonzero 𝐴∈𝐂^𝑛×𝑛. Then, 𝑓(𝐵)∈𝐶_𝐴∩𝐂^𝑛×𝑛, and equivalently 𝑣(𝑓(𝐵))∈Ker(𝐴⊕(−𝐴^∗)), for any function 𝑓∶𝐂^𝑛×𝑛→𝐂^𝑛×𝑛 of the matrix 𝐵.

The following corollaries are direct from Theorem 4.8 from the subsequent facts:

(1)𝐴∈𝐶_𝐴;∀𝐴∈𝐂^𝑛×𝑛,(2)One has [𝐴,𝐵]=0⟹[𝐴,𝑔(𝐵)]=0⟹[𝑓(𝐴),𝑔(𝐵)]=[𝑝(𝐴),𝑔(𝐵)]=^𝜇_𝑖=0=𝛼_𝑖𝐴^𝑖,𝑔(𝐵)_𝜇_𝑖=0𝛼_𝑖𝐴^𝑖−1[𝐴,𝑔(𝐵)]=0⟺𝑔(𝐵)∈𝐶_𝑓(𝐴)∩𝐂^𝑛×𝑛,(4.13) where 𝑓(𝐴)=𝑝(𝐴), from the definition of 𝑓 being a function of the matrix 𝐴, with 𝑝(𝜆) being a polynomial fulfilling 𝑝^(𝑖)(𝜆_𝑘)=𝑓^(𝑖)(𝜆_𝑘); ∀𝑘∈𝜎(𝐴), ∀𝑖∈𝑚_𝑘−1∪{0}; ∀𝑘∈𝜇 (the number of distinct elements in 𝜎(𝐴)), where 𝑚_𝑘 is the index of 𝜆_𝑘, that is, its multiplicity in the minimal polynomial of 𝐴.(3)Theorem 4.8 is extendable for any countable set {𝑓_𝑖(𝐵)} of matrix functions of 𝐵.

Corollary 4.9. Consider a nonzero matrix 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛 for any given nonzero 𝐴∈𝐂^𝑛×𝑛. Then, 𝑔(𝐵)∈𝐶_𝑓(𝐴)∩𝐂^𝑛×𝑛 for any function 𝑓∶𝐂^𝑛×𝑛→𝐂^𝑛×𝑛 of the matrix 𝐴 and any function 𝑔∶𝐂^𝑛×𝑛→𝐂^𝑛×𝑛 of the matrix 𝐵.

Corollary 4.10. 𝑓(𝐴)∈𝐶_𝐴∩𝐂^𝑛×𝑛, and equivalently 𝑣(𝑓(𝐴))∈Ker(𝐴⊕(−𝐴^∗)), for any function 𝑓∶𝐂^𝑛×𝑛→𝐂^𝑛×𝑛 of the matrix 𝐴.

Corollary 4.11. If 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛 then any countable set of function matrices {𝑓_𝑖(𝐵)} is 𝐶_𝐴 and in MC_𝐴.

Corollary 4.12. Consider any countable set of function matrices 𝐶_𝐹∶={𝑓_𝑖(𝐴);∀𝑖∈𝑝}⊂𝐶_𝐴 for any given nonzero 𝐴∈𝐂^𝑛×𝑛. Then, ⋂_{𝑓𝑖∈𝐶𝐹}(Ker(𝑓_𝑖(𝐴)⊕(−𝑓_𝑖(𝐴^∗))))⊃Ker(𝐴⊕(−𝐴^∗)).

Note that matrices which commute and are simultaneously triangularizable through the same similarity transformation maintain a zero commutator after such a transformation is performed.

Theorem 4.13. Assume that 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛. Thus, Λ_𝐵∈𝐶_Λ𝐴∩𝐂^𝑛×𝑛 provided that there exists a nonsingular matrix 𝑇∈𝐂^𝑛×𝑛 such that Λ_𝐴=𝑇⁻¹𝐴𝑇 and Λ_𝐵=𝑇⁻¹𝐵𝑇.

A direct consequence of Theorem 4.13 is that if a set of matrices are simultaneously triangularizable to their real canonical forms by a common transformation matrix then the pair wise commuting properties are identical to those of their respective Jordan forms.

Appendices

A. Proofs of the Results of Section 2

Proof of Proposition 2.1. (i)-(ii) First note by inspection that ∅≠𝐶_𝐴⊃{0,𝐴}; ∀𝐴∈𝐑^𝑛×𝑛. Also, [𝐴,𝑋]==𝐴𝐴𝑋⊕−𝑋−𝐴𝐴=_𝑇𝐴⊗𝐼_𝑛−𝐼_𝑛⊗𝐴^𝑇𝑣(𝑋)𝑣(𝑋)=0⟹𝑣(𝑋)∈Ker𝐴⊕−𝐴^𝑇(A.1) and Proposition 2.1(i)-(ii) has been proved since there is an isomorphism 𝑓∶𝐑^𝑛2↔𝐑^𝑛×𝑛 defined by 𝑓(𝑣(𝑋))=𝑋; ∀𝑋∈𝐑^𝑛×𝑛 for 𝑣(𝑋)=(𝑥^𝑇1,𝑥^𝑇2,…,𝑥^𝑇𝑛)^𝑇∈𝐑^𝑛2 if 𝑥^𝑇𝑖∶=(𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑛) is the ith row of the square matrix 𝑋.
(iii) It is a direct consequence of Proposition 2.1(iii) and the symmetry property of the commutator of two commuting matrices 𝐵∈𝐶_𝐴⇔[𝐴,𝐵]=[𝐵,𝐴]=0⇔𝐴∈𝐶_𝐵.

Proof of Proposition 2.2. [𝐴,𝐴]=0;∀𝐴∈𝐑^𝑛×𝑛⇒∃𝐑^𝑛2∋0≠𝑣(𝐴)∈Ker(𝐴⊕(−𝐴^𝑇)); ∀𝐴∈𝐑^𝑛×𝑛. As a result, Ker𝐴⊕−𝐴^𝑇≠0∈𝐑^𝑛2;∀𝐴∈𝐑^𝑛×𝑛⟺rank𝐴⊕−𝐴^𝑇<𝑛²;∀𝐴∈𝐑^𝑛×𝑛(A.2) so that 0∈𝜎(𝐴⊕(−𝐴^𝑇)).
Also, ∃𝑋(≠0)∈𝐑^𝑛×𝑛∶[𝐴,𝑋]=0⇔𝑋∈𝐶_𝐴 since Ker(𝐴⊕(−𝐴^𝑇))≠0∈𝐑^𝑛2.
Proposition 2.2 has been proved.

Proof of Theorem 2.3. (i) Note that 𝜎(𝐴)==𝜎𝜎₀𝐴^𝑇⟹𝜎𝐴⊕−𝐴^𝑇∶=𝐂∋𝜂=𝜆_𝑘−𝜆_ℓ;∀𝜆_𝑘,𝜆_ℓ∈𝜎(𝐴);∀𝑘,ℓ∈𝑛𝐴⊕−𝐴^𝑇∪𝜎₀𝐴⊕−𝐴^𝑇,(A.3) where 𝜎₀𝐴⊕−𝐴^𝑇=𝜆∈𝜎𝐴⊕−𝐴^𝑇∶𝜆=0,𝜎₀𝐴⊕−𝐴^𝑇=𝜆∈𝜎𝐴⊕−𝐴^𝑇∶𝜆≠0=𝜎𝐴⊕−𝐴^𝑇⧵𝜎₀𝐴⊕−𝐴^𝑇.(A.4) Furthermore, 𝜎(𝐴⊕(−𝐴^𝑇))∶={𝐂∋𝜆=𝜆_𝑗−𝜆_𝑖∶𝜆_𝑖,𝜆_𝑗∈𝜎(𝐴);∀𝑖,𝑗∈𝑛} and 𝑧_𝑖⊗𝑥_𝑗 is a right eigenvector of 𝐴⊕(−𝐴^𝑇) associated with its eigenvalue 𝜆_𝑗𝑖=𝜆_𝑗−𝜆_𝑖. 𝜆=𝜆_𝑗−𝜆_𝑖∈𝜎(𝐴⊕(−𝐴^𝑇)) has algebraic and geometric multiplicities 𝜇_𝑗𝑖 and 𝜈_𝑗𝑖, respectively; ∀𝑖,𝑗∈𝑛, since 𝑥_𝑗 and 𝑧_𝑖 are, respectively, the right eigenvectors of 𝐴 and 𝐴^𝑇 with associated eigenvalues 𝜆_𝑗 and 𝜆_𝑖;∀𝑖,𝑗∈𝑛.
Let 𝐽_𝐴 be the Jordan canonical form of 𝐴. It is first proved that there exists a nonsingular 𝑇∈𝐑^𝑛2×𝑛2 such that 𝐽_𝐴⊕(−𝐽_𝐴^𝑇)=𝑇⁻¹(𝐴⊕(−𝐴^𝑇))𝑇. The proof is made by direct verification by using the properties of the Kronecker product, with 𝑇=𝑃⊗𝑃^𝑇 for a nonsingular 𝑃∈𝐑^𝑛×𝑛 such that 𝐴∼𝐽_𝐴=𝑃⁻¹𝐴𝑃, as follows: 𝑇⁻¹𝐴⊕−𝐴^𝑇𝑇==𝑃𝑃⊗₋₁𝑃^𝑇⁻¹𝐴⊗𝐼_𝑛𝑃⊗𝑃^𝑇−𝑃⊗𝑃^𝑇⁻¹𝐼_𝑛⊗𝐴^𝑇𝑃⊗𝑃^𝑇=𝑃₋₁𝐴𝑃⊗𝑃^−𝑇𝐼_𝑛𝑃^𝑇−𝑃⁻¹𝐼_𝑛𝑃⊗𝑃^−𝑇𝐴^𝑇𝑃^𝑇=𝐽_𝐴𝐴𝑃⊗𝐼_𝑛−𝐼_𝑛⊗𝑃^−𝑇𝐴^𝑇𝑃^𝑇⊗𝐼_𝑛−𝐼_𝑛⊗𝐽_𝐴^𝑇=𝐽_𝐴⊗𝐼_𝑛+𝐼_𝑛⊗−𝐽_𝐴^𝑇=𝐽_𝐴⊕−𝐽_𝐴^𝑇(A.5) and the result has been proved. Thus, rank(𝐴⊕(−𝐴^𝑇))=rank(𝐽_𝐴⊕(−𝐽_𝐴^𝑇)). It turns out that 𝑃 is, furthermore, unique except for multiplication by any nonzero real constant. Otherwise, if 𝑇≠𝑃⊗𝑃^𝑇, then there would exist a nonsingular 𝑄∈𝐑^𝑛×𝑛 with 𝑄≠𝛼𝐼_𝑛;∀𝛼∈𝐑 such that 𝑇=𝑄(𝑃⊗𝑃^𝑇)⁻¹𝑄 so that 𝑇⁻¹(𝐴⊕(−𝐴^𝑇))𝑇≠𝐽_𝐴⊕(−𝐽_𝐴^𝑇) provided that 𝑃⊗𝑃^𝑇⁻¹𝐴⊕−𝐴^𝑇𝑃⊗𝑃^𝑇=𝐽_𝐴⊕−𝐽_𝐴^𝑇.(A.6) Thus, note that card𝜎𝐴⊕−𝐴^𝑇=𝑛²=^𝜇_𝑖=1𝜇_𝑖𝑖=^𝜇_𝑖=1𝜇_𝑖²≥𝜇(0)=^𝜇_𝑖=1𝜇_𝑖𝑖=^𝜇_𝑖=1≥𝜇^2𝑖≥𝜈𝜈(0)=^𝜇_𝑖=1𝜈_𝑖𝑖=^𝜇_𝑖=1𝜈^2𝑖=^𝜇_𝜇𝑖=1_𝑗=1𝜈_𝑖𝑗²−2^𝜇_𝑖=10𝑥0200𝑑^𝜇_𝑗(_≠𝑖)=1𝜈_𝑖𝑗=𝜈−2^𝜇_𝑖=10𝑥0200𝑑^𝜇_𝑗(_≠𝑖)=1𝜈_𝑖𝑗≥𝑛.(A.7) Those results follow directly from the properties of the Kronecker sum 𝐴⊕𝐵 of n-square real matrices 𝐴 and 𝐵=−𝐴^𝑇 since direct inspection leads to the following.(1)0∈𝜎(𝐴⊕(−𝐴^𝑇)) with algebraic multiplicity 𝜇(0)≥∑^𝜇𝑖=1𝜇_𝑖𝑖=∑^𝜇𝑖=1𝜇^2𝑖≥∑^𝜇𝑖=1𝜈^2𝑖≥𝑛 since there are at least ∑^𝑛𝑖=1𝜇^2𝑖 zeros in 𝜎(𝐴⊕(−𝐴^𝑇)) (i.e., the algebraic multiplicity of 0∈𝜎(𝐴⊕(−𝐴^𝑇)) is at least ∑^𝑛𝑖=1𝜇^2𝑖) and since 𝜈_𝑖≥1; ∀𝑖∈𝑛. Also, a simple computation of the number of eigenvalues of 𝐴⊕(−𝐴^𝑇) yields card𝜎(𝐴⊕(−𝐴^𝑇))=𝑛²=∑^𝜇𝑖=1𝜇_𝑖𝑖=(∑^𝜇𝑖=1𝜇_𝑖)².(2)The number of linearly independent vectors in 𝑆 is 𝜈=∑^𝜇𝑖=1∑^𝜇𝑗=1𝜈_𝑖𝑗=(∑^𝜇𝑖=1𝜈_𝑖)²≥∑^𝜇𝑖=1𝜈_𝑖𝑖=∑^𝜇𝑖=1𝜈^2𝑖 since the total number of Jordan blocks in the Jordan canonical form of 𝐴 is ∑^𝜇𝑖=1𝜈_𝑖.(3)The number of Jordan blocks associated with 0∈𝜎(𝐴⊕(−𝐴^𝑇)) in the Jordan canonical form of (𝐴⊕(−𝐴^𝑇)) is 𝜈(0)=∑^𝜇𝑖=1𝑣^2𝑖≤𝜈, with 𝜈_𝑖𝑖=𝜈^2𝑖𝑖; ∀𝑖∈𝑛. Thus, card𝜎₀𝐴⊕−𝐴^𝑇=^𝜇_𝑖=1𝜇_𝑖𝑖=^𝜇_𝑖=1𝜇^2𝑖,card𝜎₀𝐴⊕−𝐴^𝑇=𝑛²−^𝜇_𝑖=1rank𝐴⊕−𝐴_𝑇𝜇^2𝑖,=𝑛²−𝜈(0)=𝑛²−^𝜇_𝑖=1𝑣^2𝑖,dimKer𝐴⊕−𝐴^𝑇=𝜈(0)=^𝜇_𝑖=1𝑣^2𝑖.(A.8)(4)There are at least 𝜈(0) linearly independent vectors in 𝑆∶=span{𝑧_𝑖⊗𝑥_𝑗,∀𝑖,𝑗∈𝑛}. Also, the total number of Jordan blocks in the Jordan canonical form of (𝐴⊕(−𝐴^𝑇)) is 𝜈=dim𝑆=(∑^𝜇𝑖=1∑^𝜇𝑗=1𝜈_𝑖𝑗)=(∑^𝜇𝑖=1𝜈_𝑖)²=𝜈(0)+2∑^𝜇𝑖=1∑^{𝜇𝑗(≠𝑖)=1}𝜈_𝑖𝑗≥𝜈(0).
Property (i) has been proved. Property (ii) follows directly from the orthogonality in 𝐑^𝑛2of its range and null subspaces.

Proof of Theorem 2.4. First note from Proposition 2.1 that 𝑋∈𝐶_𝐴 if and only if (𝐴⊕(−𝐴^𝑇))𝑣(𝑋)=0 since 𝑣(𝑋)∈Ker(𝐴⊕(−𝐴^𝑇)). Note also from Proposition 2.1 that 𝑋∈𝐶_𝐴 if and only if 𝑣(𝑋)∈Im(𝐴⊕(−𝐴^𝑇)). Thus, 𝑋∈𝐶_𝐴 if and only if 𝑣(𝑋) is a solution to the algebraic compatible linear system: 𝐴⊕−𝐴^𝑇𝑣(𝑋)=𝑣(𝑀)(A.9) for any 𝑀(≠0)∈𝐑^𝑛×𝑛 such that rank𝐴⊕−𝐴^𝑇=rank𝐴⊕−𝐴^𝑇,𝑣(𝑀)=𝑛²−𝜈(0).(A.10) From Theorem 2.3, the nullity and the rank of 𝐴⊕(−𝐴^𝑇) are, respectively, dimKer(𝐴⊕(−𝐴^𝑇))=𝜈(0)rank(𝐴⊕(−𝐴^𝑇))=𝑛²−𝜈(0). Therefore, there exist permutation matrices 𝐸,𝐹∈𝐑^𝑛2×𝑛2 such that there exists an equivalence transformation: 𝐴⊕−𝐴^𝑇≈𝐴∶=𝐸𝐴⊕−𝐴^𝑇𝐹=Blockmatrix𝐴_𝑖𝑗;𝑖,𝑗∈2(A.11) such that 𝐴₁₁ is square nonsingular and of order 𝜈₀. Define 𝑀=𝐸𝑀𝐹≈𝑀(≠0)∈𝐑^𝑛×𝑛. Then, the linear algebraic systems (𝐴⊕(−𝐴^𝑇))𝑣(𝑋)=𝑣(𝑀), and 𝐸𝐴⊕−𝐴^𝑇𝐹𝑣𝑋=𝐴₁₁𝐴₁₂𝐴₂₁𝐴₂₂⎡⎢⎢⎣𝑣𝑣𝑋₁𝑋₂⎤⎥⎥⎦=⎡⎢⎢⎣𝑣𝑣𝑀₁𝑉𝑀₂⎤⎥⎥⎦,𝑋₁⟺=𝐴⁻¹¹¹𝑣𝑀₁−𝐴₁₂𝑣𝑋₂𝐴₂₂−𝐴₂₁𝐴⁻¹¹¹𝐴₁₂𝑉𝑋₂=𝑣𝑀₂−𝐴₂₁𝐴⁻¹¹¹𝑣𝑀₁(A.12) are identical if 𝑋 and 𝑀 are defined according to 𝑣(𝑋)=𝐹𝑣(𝑋) and 𝑣(𝑀)=𝐸𝑣(𝑀). As a result, Properties (i) and (ii) follow directly from (A.12) for 𝑀=𝑀=0 and for any 𝑀 satisfying rank(𝐴⊕(−𝐴^𝑇))=rank(𝐴⊕(−𝐴^𝑇),𝑣(𝑀))=𝑛²−𝜈(0), respectively.

B. Proofs of the Results of Section 3

Proof of Proposition 3.1. (i) The first part of Property (i) follows directly from Proposition 2.1 since all the matrices of 𝐀_𝐶 pair wise commute and any arbitrary matrix commutes with itself (thus 𝑗=𝑖 may be removed from the intersections of kernels of the first double sense implication). The last part of Property (i) follows from the antisymmetric property of the commutator [𝐴_𝑖,𝐴_𝑗]=[𝐴_𝑗,𝐴_𝑖]=0;∀𝐴_𝑖,𝐴_𝑗∈𝐀_𝐶 what implies 𝐴_𝑖∈𝐀_𝐶;∀𝑖∈𝑝⇔𝑣(𝐴_𝑖)∈⋂_{𝑖+1≤𝑗≤𝑝}Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗));∀𝐴_𝑖,𝐴_𝑗∈𝐀_𝐶.
(ii) It follows from its equivalence with Property (i) since Ker𝑁_𝑖(𝐀_𝐶)≡⋂_{𝑗(≠𝑖)∈𝑝}Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗)).
(iii) Property (iii) is similar to Property (i) for the whose set 𝑀_𝐀𝐶 of matrices which commute with the set 𝐀_𝐶 so that it contains 𝐀_𝐶 and, furthermore, Ker𝑁(𝐀_𝐶)≡⋂_𝑖∈𝑝Ker(𝐴_𝑖⊕(−𝐴^𝑇𝑖)).
(iv) It follows from ⋃_𝑗∈𝑝Im(𝐴_𝑗⊕(−𝐴^𝑇𝑗))=⋂_𝑗∈𝑝Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗));𝐴_𝑗∈𝐀_𝐶 and 𝐑^𝑛2∋0∈Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗))∩Im(𝐴_𝑗⊕(−𝐴^𝑇𝑗)) but 𝐑^𝑛×𝑛∋𝑋=0 commutes with any matrix in 𝐑^𝑛×𝑛 so that 𝐑^𝑛×𝑛∋0∈MC_𝐀𝐶⇔𝐑^𝑛×𝑛∋0∉MC_𝐀𝐶 for any given 𝐀_𝐶.
(v) and (vi) are similar to (ii)–(iv) except that the members of 𝐀 do not necessarily commute.

Proof of Proposition 3.2. It is a direct consequence from Proposition 3.1(i)-(ii) since the existence of nonzero pair wise commuting matrices (all the members of 𝐀_𝐶) implies that the above matrices 𝑁(𝐀_𝐶),𝑁_𝑖(𝐀_𝐶),𝐴_𝑗⊕(−𝐴^𝑇𝑗) are all rank defective and have at least identical number of rows than that of columns. Therefore, the square matrices 𝑁^𝑇(𝐀_𝐶)𝑁(𝐀_𝐶),𝑁^𝑇𝑖(𝐀_𝐶)𝑁_𝑖(𝐀_𝐶), and 𝐴_𝑗⊕(−𝐴^𝑇𝑗) are all singular.

Proof of Theorem 3.3. (i) Any nonzero matrix Λ=diag(𝜆𝜆⋯𝜆), 𝜆(≠0)∈𝐑 is such that Λ(≠0)∈𝐶_𝐴𝑖(∀𝑖∈𝑝) so that Λ∈𝐶_𝐀. Thus, 0≠𝑣(Λ)∈Ker𝑁(𝐀)⇔𝑛²>𝑟𝑎𝑛𝑘𝑁(𝐀)≥rank𝑁_𝑖(𝐀)≥rank(𝐴_𝑖⊕(−𝐴_𝑖)); ∀𝑖∈𝑝 and any given set 𝐀. Property (i) has been proved.
(ii) The first part follows by contradiction. Assume ⋂_𝑖∈𝑝Ker(𝐴_𝑖⊕(−𝐴^𝑇𝑖))={0} then 0≠𝑣(Λ)∉Ker𝑁(𝐀) so that Λ=diag(𝜆𝜆⋯𝜆)∉𝐶_𝐀, for any 𝜆(≠0)∈𝐑 what contradicts (i). Also, 𝑋∈𝐶_𝐴𝑖⇔𝑣(𝑋)∈Ker(𝐴_𝑖⊕(−𝐴^𝑇𝑖)); ∀𝑖∈𝑝 so that 𝑋∈𝐶_𝐀⇔𝑣(𝑋)∈⋂_𝑖∈𝑝Ker(𝐴_𝑖⊕(−𝐴^𝑇𝑖)) what is equivalent to its contrapositive logic proposition 𝑋∈𝐶_𝐀⇔𝑣(𝑋)∈⋃_𝑖∈𝑝Im(𝐴_𝑖⊕(−𝐴^𝑇𝑖)).
(iii) Let 𝐀=𝐀_𝐶⇔𝐴_𝑖∈𝐶_𝐴𝑗;∀𝑗(≠𝑖)∈𝑝,∀𝑖∈𝑝⇔𝐴_𝑖∈𝐶_𝐴𝑗;∀𝑗,𝑖∈𝑝 since 𝐴_𝑖∈𝐶_𝐴𝑖;∀𝑖∈𝑝𝑣𝐴_𝑖∈_𝑖∈𝑝Ker𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝑖∈𝑝⟺𝑣𝐴_𝑖∈_{𝑖∈𝑝⧵{𝑖}}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗;∀𝑖∈𝑝.(B.1) On the other hand, ⎛⎜⎜⎝𝑣𝐴_𝑖∈_{𝑗∈𝑝⧵𝑖}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗⟺𝑣𝐴_𝑖∈𝐶_𝐴𝑗;∀𝑗∈𝑝⎞⎟⎟⎠forany𝑖(<𝑝)∈𝑝.(B.2) This assumption implies directly that 𝑣𝐴_𝑖∈𝐶_𝐴𝑗;∀𝑗∈𝑝∧𝑣𝐴_𝑖+1∈_{𝑗∈𝑖+1}𝐶_𝐴𝑗forany𝑖(<𝑝)∈𝑝(B.3) which together with 𝑣(𝐴_𝑖+1)∈⋂_{𝑗∈𝑝⧵𝑖+1}Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗)) implies that 𝑣𝐴_𝑖+1∈𝐶_𝐴𝑗;∀𝑗∈𝑝⟹⎛⎜⎜⎝𝑣𝐴_𝑖+1∈_{𝑗∈𝑝⧵𝑖+1}Ker𝐴_𝑗⊕−𝐴^𝑇𝑗⎞⎟⎟⎠for(𝑖+1)∈𝑝.(B.4) Thus, it follows by complete induction that 𝐀=𝐀_𝐶⇔𝑣(𝐴_𝑖)∈⋂_{𝑖∈𝑝⧵{𝑖}}Ker(𝐴_𝑗⊕(−𝐴^𝑇𝑗));∀𝑖∈𝑝 and Property (iii) has been proved.
(iv) The definition of M𝐀_𝐶 follows from Property (iii) in order to guarantee that [𝑋,𝐴_𝑖]=0; ∀𝐴_𝑖∈𝐀. The fact that such a set contains properly 𝐀_𝐶∪{0} follows directly from 𝐑^𝑛×𝑛∋Λ=diag(𝜆𝜆⋯𝜆)(∈MC_𝐀𝐶)≠𝐀_𝐶∪{0} for any 𝐑∋𝜆≠0.

Proof of Theorem 3.4. If 𝐴_𝑖∼𝐽_𝐴𝑖=𝑃^𝑖−1𝐴_𝑖𝑃_𝑖, with 𝐽_𝐴𝑖 being the Jordan canonical form of 𝐴_𝑖 then 𝐴_𝑖⊕(−𝐴^𝑇𝑖)∼𝐽_𝐴𝑖⊕(−𝐽^𝑇𝐴_𝑖)=𝑇^𝑖−1(𝐴_𝑖⊕(−𝐴^𝑇𝑖))𝑇_𝑖 with 𝑇_𝑖=𝑃_𝑖⊗𝑃^𝑇𝑖∈𝐑^𝑛2×𝐑^𝑛2 (see proof of Theorem 2.3) being nonsingular; ∀𝑖∈𝑝. Thus, (𝐴_𝑖⊕(−𝐴^𝑇𝑖))=𝑇_𝑖(𝐽_𝐴𝑖⊕(−𝐽^𝑇𝐴_𝑖))𝑇^𝑖−1 so that: 𝑁(𝐀)=𝐴^𝑇1⊕−𝐴₁𝐴^𝑇2⊕−𝐴₂⋯𝐴^𝑇𝑝⊕−𝐴_𝑝^𝑇=𝐼_𝑝𝑛²−𝑈𝑊^𝑇1𝑊^𝑇2^𝑇=𝐓𝐉𝐓_𝐚,(B.5) where 𝐓𝐓_𝐚∶=BlockDiag𝑇₁𝑇₂⋯𝑇_𝑝∈𝐑^{𝑝𝑛2×𝑝𝑛2},𝐉∶∶==B𝑇loc^1−𝑇k𝑇Dia^2−𝑇g⋯𝐽_𝐴1𝑇^𝑝−𝑇^𝑇∈𝐑^{𝑝𝑛2×𝑛2},⊕−𝐽^𝑇𝐴₁𝐽^𝐴₂⊕−𝐽^𝑇𝐴₂⋯𝐽^𝐴_𝑝⊕−𝐽^𝑇𝐴_𝑝∈𝐑^{𝑝𝑛2×𝑝𝑛2}.(B.6) Then, Ker𝑁(𝐀)=^𝑝_𝑖=1≡_𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖=Ker𝐉𝐓_𝐚_𝑖=1Ker𝐽^𝐴_𝑖⊕−𝐽^𝑇𝐴_𝑖𝑃^𝑖−1⊗𝑃^𝑖−𝑇(B.7) since 𝐓 is nonsingular. Thus, ∀𝑋∈Dom(𝐀)⊂𝐑^𝑛2: 𝑋∈𝐶_𝐀⟺⟺𝑣𝑣((𝑋𝑋))∈∈Ker_𝑝𝑁(𝐀)_𝑖=1⟺𝑣(𝑋)∈_𝑝Ker𝐴_𝑖⊕−𝐴^𝑇𝑖_𝑖=1⟺𝑣(𝑋)∈Im𝑃K_𝑖er𝐽_𝐴𝑖⊕−𝐽^𝑇𝐴_𝑖𝑃^𝑖−1⊗𝑃^𝑖−𝑇⟺𝑣(𝑋)∈_𝑝⊗𝑃^𝑖−1Ker𝐽_𝐴𝑖⊕−𝐽^𝑇𝐴_𝑖;∀𝑖∈𝑝_𝑖=1Im𝑃_𝑖⟺𝑣(𝑋)∈_𝑝⊗𝑃^𝑖−1(𝑌)_𝑖=1Im𝑃_𝑖⊗𝑃^𝑖−1𝑌_𝑖,(B.8) where 𝑌_𝑖∈Ker(𝐽_𝐴𝑖⊕(−𝐽^𝑇𝐴_𝑖)); ∀𝑖∈𝑝 and 𝑌∈(⋂^𝑝𝑖=1(Ker(𝐽_𝐴𝑖⊕(−𝐽^𝑇𝐴_𝑖)))). Property (i) has been proved. The first inequality of Property (ii) follows directly from Property (i). The results of equalities and inequalities in the second line of Property (ii) follow by the first inequality by taking into account Theorem 2.3. Property (iii) follows from the proved equivalent definitions of 𝐶_𝐀 in Property (i) by taking into account that [𝐴_𝑗,𝐴_𝑗]=0; ∀𝑗∈𝑝 so that 𝑣𝐴_𝑗∈^𝑝_𝑖=1Ker𝐽^𝐴_𝑖⟺𝑣𝐴_𝑗⊕−𝐽^𝑇𝐴_𝑖𝑃^𝑖−1⊗𝑃^𝑖−𝑇∈^𝑝_{𝑖(≠𝑗)=1}Ker𝐽^𝐴_𝑖⊕−𝐽^𝑇𝐴_𝑖𝑃^𝑖−1⊗𝑃^𝑖−𝑇;∀𝑗∈𝑝.(B.9)

C. Proofs of the Results of Section 4

Proofs of Propositions 4.1–4.3Proposition 4.1 follows by inspection of (4.1). Proposition 4.2 implies that Proposition 4.1 holds with the four involved commutators being zero. Then the left condition of Proposition 4.2 implies that 𝐵∈𝐶_𝐴, from Proposition 4.1, so that Proposition 4.2 holds. Proposition 4.3 is equivalent to Proposition 4.2.

Proof of Theorem 4.4. (i) Equation (4.2) is a rearrangement in an equivalent algebraic system of Proposition 4.1 in the unknown 𝑣(𝐵_im) for given 𝐴 and 𝐵_re. The system is compatible if (4.2) holds from the Kronecker-Capelli theorem. The proof of Property (ii) is similar to that of (i) with the appropriate interchange of roles of 𝐵_re and 𝐵_im.
(iii) Since rank𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re<𝑛²(C.1) from Theorem 3.3(i) then 0≠𝐵=𝐵_re∈𝐶_𝐴 if and only if 𝐵_re∈𝐶_𝐴re∩𝐶_𝐴im(≠∅)⊂𝐶_𝐴. The same proof follows for 0≠𝐵=𝐵_im∈𝐶_𝐴 since rank𝐴^re𝐴_im⊕−𝐴^𝑇_re⊕−𝐴^𝑇_im=rank𝐴^im𝐴_re⊕−𝐴^𝑇_im⊕−𝐴^𝑇_re<𝑛².(C.2)

Proof of Theorem 4.5. (i) It follows in the same way as that of Theorem 4.4 by rewriting the algebraic system (4.3) in the form (4.5) which has nonzero solutions if (4.8) holds. But (4.8) always holds since 𝐵=𝐴∈𝐶_𝐴∩𝐂^𝑛×𝑛 is nonzero if 𝐴 is nonzero and if 𝐴=0∈𝐂^𝑛×𝑛 then 𝐶_𝐴=𝐂^𝑛×𝑛.
(ii) Direct calculations yield the equivalence of (4.5) with the separation into real and imaginary parts of the subsequent algebraic system: 𝐴⊗𝐼_𝑛−𝐼_𝑛⊗𝐴^∗𝑣(𝐵)=𝐴_re+𝐢𝐴_im⊗𝐼_𝑛−𝐼_𝑛⊗𝐴^𝑇_re−𝐢𝐴^𝑇_im𝑣𝐵_re+𝐢𝑣𝐵_im=0(C.3) which is always solvable with a nonzero solution (i.e., compatible) since rank(𝐴⊗𝐼_𝑛−𝐼_𝑛⊗𝐴^∗)<𝑛² (otherwise, 𝐴(≠0)∈𝐶_𝐴).

Outline of Proof of Theorem 4.6
(i) It is a direct extension of Theorem 4.5 by decomposing the involved complex matrices in their real and imaginary parts since from Theorem 3.3(i) both left block matrices in the coefficient matrix of (4.11) have rank less than 𝑛². As a result, such a coefficient matrix has rank less than 2𝑛² so that nonzero solutions exists to the algebraically compatible system of linear equations (4.11). As a result, a nonzero 𝑛-square complex commuting matrix exists.
(ii) It is close to that of (i) but the rank condition for compatibility of the algebraic system is not needed since the coefficient matrix of (4.11) is rank defective since 𝐴_𝑗∈𝐀_𝐶⇔(𝑣^𝑇(𝐴_𝑗re),𝑣^𝑇(𝐴_𝑗im))^𝑇 is in the null space of the coefficient matrix; ∀𝑗∈𝑝.
(iii) Its proof is close to that of Theorem 4.5(ii) and it is then omitted.

Proof of Theorem 4.8. For any 𝐵∈𝐶_𝐴∩𝐂^𝑛×𝑛: [𝐴,𝐵]⟹=0⟹𝜆𝐼_𝑛𝜆𝐼_𝑛−𝐵𝐴=𝐴𝜆𝐼_𝑛−𝐵;∀𝜆∈𝐂⟹[𝐴,𝑓−(𝐵𝐵)]=⁻¹𝐴𝐴=𝐴1𝜆𝐼_𝑛−𝐵⁻¹;∀𝜆∈𝐂∩𝜎(𝐵)2𝜋𝐢_𝐶𝑓(𝜆)𝜆𝐼_𝑛−𝐵⁻¹𝑑𝜆=12𝜋𝐢_𝐶=1𝑓(𝜆)𝜆𝐼_𝑛−𝐵⁻¹𝐴𝑑𝜆2𝜋𝐢_𝐶𝑓(𝜆)𝜆𝐼_𝑛−𝐵⁻¹𝑑𝜆𝐴=[𝑓(𝐵),𝐴]=0,(C.4) where 𝐶 is the boundary of 𝐷 and consists in a set of closed rectifiable Jordan curves which contains no point of 𝜎(𝐴) since 𝜆∈𝐂∩𝜎(𝐴) so that the identity (𝜆𝐼_𝑛−𝐵)⁻¹𝐴=𝐴(𝜆𝐼_𝑛−𝐵)⁻¹ is true. Then, 𝑓(𝐵)∈𝐶_𝐴∩𝐂^𝑛×𝑛 has been proved. From Theorem 4.5, this is equivalent to 𝑣(𝑓(𝐵))∈Ker(𝐴⊕(−𝐴^∗)).

Proof of Theorem 4.13. 𝐵∈𝐶_𝐴⇔𝐹[𝐴,𝐵]𝐺=0;∀𝐹,𝐺∈𝐂^𝑛×𝑛 being nonsingular. By choosing 𝐹⁻¹=𝐺=𝑇, it follows that 𝑇⁻¹[𝐴,𝐵]𝑇=𝑇⁻¹𝐴𝑇𝑇⁻¹𝐵𝑇−𝑇⁻¹𝐵𝑇𝑇⁻¹𝐴𝑇=Λ_𝐴,Λ_𝐵=0.(C.5)

Acknowledgments

The author is grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2006-00714. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07 and SAIOTEK S-PE08UN15. Finally, he is grateful to the reviewers by their interesting suggestions.