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Mathematical Problems in Engineering
Volumeย 2009ย (2009), Article IDย 650970, 24 pages
Research Article

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

Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia). Aptdo. 644, 48080 Bilbao, Spain

Received 5 January 2009; Revised 4 May 2009; Accepted 9 June 2009

Academic Editor: Angeloย Luongo

Copyright ยฉ 2009 M. De La Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper investigates the necessary and sufficient condition for a set of (real or complex) matrices to commute. It is proved that the commutator [๐ด,๐ต]=0 for two matrices ๐ด and ๐ต if and only if a vector ๐‘ฃ(๐ต) defined uniquely from the matrix ๐ต is in the null space of a well-structured matrix defined as the Kronecker sum ๐ดโŠ•(โˆ’๐ดโˆ—), 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 ๐‘“(๐ด) which extend the well-known sufficiency-type commuting result [๐ด,๐‘“(๐ด)]=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 [1โ€“22] including Fourier transform theory, graph theory where, for instance, the commutativity of the adjacency matrices is relevant [1, 17โ€“19, 21โ€“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., [23โ€“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 [36โ€“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 [23โ€“25, 28โ€“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, [10โ€“15, 27โ€“30, 34โ€“41] and references therein. The set of involved problems is wide enough like, for instance, switched multimodel techniques [27โ€“30, 35, 36, 40, 41], switched multiestimation techniques with incorporated parallel multiestimation schemes involving adaptive control [34, 38โ€“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, 27โ€“30, 34โ€“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 DFTโ€™s and DCTโ€™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 ๐ถ0-semigroup. This leads to a well-known result in Systems Theory establishing that the matrix function ๐‘’๐ด1๐‘ก1+๐ด2๐‘ก2=๐‘’๐ด1๐‘ก1๐‘’๐ด2๐‘ก2 is a fundamental (or state transition) matrix for the cascade of the time invariant differential systems ฬ‡๐‘ฅ1(๐‘ก)=๐ด1๐‘ฅ1(๐‘ก), operating on a time ๐‘ก1, and ฬ‡๐‘ฅ2(๐‘ก)=๐ด2๐‘ฅ2(๐‘ก), operating on a time ๐‘ก2, provided that ๐ด1 and ๐ด2 commute (see, e.g., [7โ€“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

[๐ด,๐ต] is the commutator of the square matrices ๐ด and ๐ต.

๐ดโŠ—๐ตโˆถ=(๐‘Ž๐‘–๐‘—๐ต) is the Kronecker (or direct) product of ๐ดโˆถ=(๐‘Ž๐‘–๐‘—) and ๐ต.

๐ดโŠ•๐ตโˆถ=๐ดโŠ—๐ผ๐‘›+๐ผ๐‘›โŠ—๐ต is the Kronecker sum of the square matrices ๐ดโˆถ=(๐‘Ž๐‘–๐‘—) and both of order ๐‘›, where ๐ผ๐‘› is the nth identity matrix.

๐ด๐‘‡ is the transpose of the matrix ๐ด and ๐ดโˆ— is the conjugate transpose of the complex matrix ๐ด. For any matrix ๐ด, Im๐ด and Ker๐ด are its associate range (or image) subspace and null space, respectively. Also, rank(๐ด) is the rank of ๐ด which is the dimension of Im(๐ด) and det(๐ด) is the determinant of the square matrix ๐ด.

๐‘ฃ(๐ด)=(๐‘Ž๐‘‡1,๐‘Ž๐‘‡2,โ€ฆ,๐‘Ž๐‘‡๐‘›)๐‘‡โˆˆ๐‚๐‘›2 if ๐‘Ž๐‘‡๐‘–โˆถ=(๐‘Ž๐‘–1,๐‘Ž๐‘–2,โ€ฆ,๐‘Ž๐‘–๐‘›) is the ith row of the square matrix ๐ด.

๐œŽ(๐ด) is the spectrum of ๐ด;๐‘›โˆถ={1,2,โ€ฆ,๐‘›}. If ๐œ†๐‘–โˆˆ๐œŽ(๐ด) then there exist positive integers ๐œ‡๐‘– and ๐œˆ๐‘–โ‰ค๐œ‡๐‘– which are, respectively, its algebraic and geometric multiplicity; that is, the times it is repeated in the characteristic polynomial of ๐ด and the number of its associate Jordan blocks, respectively. The integer ๐œ‡โ‰ค๐‘› is the number of distinct eigenvalues and the integer ๐‘š๐‘–, subject to 1โ‰ค๐‘š๐‘–โ‰ค๐œ‡๐‘–, is the index of ๐œ†๐‘–โˆˆ๐œŽ(๐ด); โˆ€๐‘–โˆˆ๐œ‡, that is, its multiplicity in the minimal polynomial of ๐ด.

๐ดโˆผ๐ต denotes a similarity transformation from ๐ด to ๐ต=๐‘‡โˆ’1๐ด๐‘‡ for given ๐ด,๐ตโˆˆ๐‘๐‘›ร—๐‘› for some nonsingular ๐‘‡โˆˆ๐‘๐‘›ร—๐‘›. ๐ดโ‰ˆ๐ต=๐ธ๐ด๐น means that there is an equivalence transformation for given ๐ด,๐ตโˆˆ๐‘๐‘›ร—๐‘› for some nonsingular ๐ธ,๐นโˆˆ๐‘๐‘›ร—๐‘›.

A linear transformation from ๐‘๐‘› to ๐‘๐‘›, represented by the matrix ๐‘‡โˆˆ๐‘๐‘›ร—๐‘›, is denoted identically to such a matrix in order to simplify the notation. If ๐‘‰โ‰ Dom๐‘‡โ‰ก๐‘๐‘› is a subspace of ๐‘๐‘› then Im๐‘‡(๐‘‰)โˆถ={๐‘‡๐‘งโˆถ๐‘งโˆˆ๐‘‰} and Ker๐‘‡(๐‘‰)โˆถ={๐‘งโˆˆ๐‘‰โˆถ๐‘‡๐‘ง=0โˆˆ๐‘๐‘›}. If ๐‘‰โ‰ก๐‘๐‘›, the notation is simplified to Im๐‘‡โˆถ={๐‘‡๐‘งโˆถ๐‘งโˆˆ๐‘๐‘›} and Ker๐‘‡โˆถ={๐‘งโˆˆ๐‘๐‘›โˆถ๐‘‡๐‘ง=0โˆˆ๐‘๐‘›}.

The symbols โ€œโ‹€โ€™โ€™ and โ€œโˆจโ€™โ€™ stand for logic conjunction and disjunction, respectively. The abbreviation โ€œiffโ€™โ€™ stands for โ€œif and only if.โ€™โ€™ The notation card ๐‘ˆ stands for the cardinal of the set ๐‘ˆ. ๐ถ๐ด (resp., ๐ถ๐ด) is the set of matrices which commute (resp., do not commute) with a matrix ๐ด. ๐ถ๐€ (resp., ๐ถ๐€) is the set of matrices which commute (resp., do not commute) with all square matrix ๐ด๐‘– belonging to a given set ๐€.

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

Consider the sets ๐ถ๐ดโˆถ={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐ด,๐‘‹]=0}โ‰ โˆ…, of matrices which commute with A, and ๐ถ๐ดโˆถ={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐ด,๐‘‹]โ‰ 0}, of matrices which do not commute with ๐ด; โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›. Note that 0โˆˆ๐‘๐‘›ร—๐‘›โˆฉ๐ถ๐ด; that is, the zero n-matrix commutes with any n-matrix so that, equivalently, 0โˆ‰๐‘๐‘›ร—๐‘›โˆฉ๐ถ๐ด and then ๐ถ๐ดโˆฉ๐ถ๐ด=โˆ…; โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›. The subsequent two basic results which follow are concerned with commutation and noncommutation of two real matrices ๐ด and ๐‘‹. The used tool relies on the calculation of the null space and the range space of the Kronecker sum of the matrix ๐ด, one of the matrices, with its minus transpose. A vector built with all the entries of the other matrix ๐‘‹ has to belong to one of the above spaces for ๐ด and ๐‘‹ to commute and to the other one in order that ๐ด and ๐‘‹ not to be two commuting matrices.

Proposition 2.1. (i) ๐ถ๐ด={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆKer(๐ดโŠ•(โˆ’๐ด๐‘‡))}.
(ii) ๐ถ๐ด=๐‘๐‘›ร—๐‘›โงตCA={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆ‰Ker(๐ดโŠ•(โˆ’๐ด๐‘‡))}โ‰ก{๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆIm(๐ดโŠ•(โˆ’๐ด๐‘‡))}.
(iii) ๐ตโˆˆ๐ถ๐ดโˆถ={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆKer(๐ดโŠ•(โˆ’๐ด๐‘‡))}.

Note that according to Proposition 2.1 the set of matrices ๐ถ๐ด which commute with the square matrix ๐ด and its complementary ๐ถ๐ด (i.e., the set of matrices which do not commute with ๐ด) can be redefined in an equivalent way by using their given expanded vector forms.

Proposition 2.2. One has ๎€ท๎€ทrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ<๐‘›2๎€ท๎€ทโŸบKer๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธโ‰ 0โŸบ0โˆˆ๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธโŸบโˆƒ๐‘‹(โ‰ 0)โˆˆ๐ถ๐ด,โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›.(2.1)

Proof. One has [๐ด,๐ด]=0;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›โ‡’โˆƒ๐‘๐‘›2โˆ‹0โ‰ ๐‘ฃ(๐ด)โˆˆKer(๐ดโŠ•(โˆ’๐ด๐‘‡)); โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›. As a result, ๎€ท๎€ทKer๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธโ‰ 0โˆˆ๐‘๐‘›2;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›๎€ท๎€ทโŸบrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ<๐‘›2;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›(2.2) so that 0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)).
Also, โˆƒ๐‘‹(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐ด,๐‘‹]=0โ‡”๐‘‹โˆˆ๐ถ๐ด since Ker(๐ดโŠ•(โˆ’๐ด๐‘‡))โ‰ 0โˆˆ๐‘๐‘›2.
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 ๐ดโŠ•(โˆ’๐ด๐‘‡).

Theorem 2.3. The following properties hold.
(i) The spectrum of ๐ดโŠ•(โˆ’๐ด๐‘‡) is ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡))={๐œ†๐‘–๐‘—=๐œ†๐‘–โˆ’๐œ†๐‘—โˆถ๐œ†๐‘–,๐œ†๐‘—โˆˆ๐œŽ(๐ด);โˆ€๐‘–,๐‘—โˆˆ๐‘›} and possesses ๐œˆ Jordan blocks in its Jordan canonical form of, subject to the constraints ๐‘›2โ‰ฅโˆ‘๐œˆ=dim๐‘†=(๐œ‡๐‘–=1๐œˆ๐‘–)2โ‰ฅ๐œˆ(0), and 0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) with an algebraic multiplicity ๐œ‡(0) and with a geometric multiplicity ๐œˆ(0) subject to the constraints: ๐‘›2=๎ƒฉ๐œ‡๎“๐‘–=1๐œ‡๐‘–๎ƒช2โ‰ฅ๐œ‡(0)โ‰ฅ๐œ‡๎“๐‘–=1๐œ‡2๐‘–โ‰ฅ๐œˆ(0)=๐œ‡๎“๐‘–=1๐œˆ2๐‘–โ‰ฅ๐‘›,(2.3) where
(a)๐‘†โˆถ=span{๐‘ง๐‘–โŠ—๐‘ฅ๐‘—,โˆ€๐‘–,๐‘—โˆˆ๐‘›}, ๐œ‡๐‘–=๐œ‡(๐œ†๐‘–) and ๐œˆ๐‘–=๐œˆ(๐œ†๐‘–) are, respectively, the algebraic and the geometric multiplicities of ๐œ†๐‘–โˆˆ๐œŽ(๐ด), โˆ€๐‘–โˆˆ๐‘›; ๐œ‡โ‰ค๐‘› is the number of distinct ๐œ†๐‘–โˆˆ๐œŽ(๐ด)(๐‘–โˆˆ๐œ‡), ๐œ‡๐‘–=๐œ‡(๐œ†๐‘–๐‘—) and ๐œˆ๐‘–๐‘—=๐œˆ(๐œ†๐‘–๐‘—), are, respectively, the algebraic and the geometric multiplicity of ๐œ†๐‘–๐‘—=(๐œ†๐‘–โˆ’๐œ†๐‘—)โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)), โˆ€๐‘–,๐‘—โˆˆ๐‘›; ๐œ‡โ‰ค๐‘›,(b)๐‘ฅ๐‘— and ๐‘ง๐‘– are, respectively, the right eigenvectors of ๐ด and ๐ด๐‘‡ with respective associated eigenvalues ๐œ†๐‘— and ๐œ†๐‘–;โˆ€๐‘–,๐‘—โˆˆ๐‘›.
(ii) One has ๎€ท๎€ทdimIm๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธ=rank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ=๐‘›2โˆ’๎€ท๎€ท๐œˆ(0)โŸบdimKer๐ดโŠ•โˆ’๐ด๐‘‡=๎€ธ๎€ธ๐œˆ(0);โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›.(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 ๐‘‹โˆˆ๐ถ๐ด๎€ท๎€ทi๏ฌ€๐ดโŠ•โˆ’๐ด๐‘‡๐‘ฃ๎€ธ๎€ธ(๐‘‹)=0โŸบ๐‘‹โˆˆ๐ถ๐ด๎‚€๐‘ฃi๏ฌ€๐‘ฃ(๐‘‹)=โˆ’๐น๐‘‡๎‚€๐‘‹2๎‚๐ด๐‘‡12๐ดโˆ’๐‘‡11,๐‘ฃ๐‘‡๎‚€๐‘‹2๎‚๎‚๐‘‡(2.5) for any ๐‘ฃ(๐‘‹2)โˆˆKer(๐ด22โˆ’๐ด21๐ดโˆ’111๐ด12), where ๐ธ,๐นโˆˆ๐‘๐‘›2ร—๐‘›2 are permutation matrices and ๐‘‹โˆˆ๐‘๐‘›ร—๐‘› and ๐‘ฃ(๐‘‹)โˆˆ๐‘๐‘›2 are defined as follows.
(a)One has ๐‘ฃ๎‚€๐‘‹๎‚โˆถ=๐นโˆ’1๎€ท๐‘ฃ(๐‘‹),๐ดโŠ•โˆ’๐ด๐‘‡๎€ธโ‰ˆ๎€ท๎€ท๐ดโˆถ=๐ธ๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ๐น,โˆ€๐‘‹โˆˆ๐ถ๐ด,(2.6) where ๐‘ฃ(๐‘‹)=(๐‘ฃ๐‘‡(๐‘‹1),๐‘ฃ๐‘‡(๐‘‹2))๐‘‡โˆˆ๐‘๐‘›2 with ๐‘ฃ(๐‘‹1)โˆˆ๐‘๐œˆ(0) and ๐‘ฃ(๐‘‹2)โˆˆ๐‘๐‘›2โˆ’๐œˆ(0).(b)The matrix ๐ด11โˆˆ๐‘๐œˆ(0)ร—๐œˆ(0) is nonsingular in the block matrix partition ๐ดโˆถ=Blockmatrix(๐ด๐‘–๐‘—;๐‘–,๐‘—โˆˆ2) with ๐ด12โˆˆ๐‘๐œˆ(0)ร—๐‘›2, ๐ด21โˆˆ๐‘(๐‘›2โˆ’๐œˆ(0))ร—๐œˆ(0) and ๐ด22โˆˆ๐‘(๐‘›2โˆ’๐œˆ(0))ร—(๐‘›2โˆ’๐œˆ(0)).
(ii) ๐‘‹โˆˆ๐ถ๐ด, for any given ๐ด(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘›, if and only if ๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๐‘ฃ๎€ธ๎€ธ(๐‘‹)=๐‘ฃ(๐‘€)(2.7) for some ๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘› such that ๎€ท๎€ทrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธ=rank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ,๐‘ฃ(๐‘€)=๐‘›2โˆ’๐œˆ(0).(2.8) Also, ๐ถ๐ด๎€ฝโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๐‘ฃ๎€ธ๎€ธ(๐‘‹)=๐‘ฃ(๐‘€)forany๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘›๎€ท๎€ทsatisfyingrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธ=rank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ,๐‘ฃ(๐‘€)=๐‘›2โˆ’๎€พ.๐œˆ(0)(2.9) Also, with the same definitions of ๐ธ, ๐น, and ๐‘‹ in (i), ๐‘‹โˆˆ๐ถ๐ด if and only if ๎‚€๐‘ฃ๐‘ฃ(๐‘‹)=๐น๐‘‡๎‚€๐‘€1๎‚๐ดโˆ’๐‘‡11โˆ’๐‘ฃ๐‘‡๎‚€๐‘‹2๎‚๐ด๐‘‡12๐ดโˆ’๐‘‡11,๐‘ฃ๐‘‡๎‚€๐‘‹2๎‚๎‚๐‘‡,(2.10) where ๐‘ฃ(๐‘‹2) is any solution of the compatible algebraic system ๎‚€๐ด22โˆ’๐ด21๐ดโˆ’111๐ด12๎‚๐‘ฃ๎‚€๐‘‹2๎‚๎‚€=๐‘ฃ๐‘€2๎‚โˆ’๐ด21๐ดโˆ’111๐‘ฃ๎‚€๐‘€1๎‚(2.11) for some ๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘› such that ๐‘‹,๐‘€โˆˆ๐‘๐‘›ร—๐‘› which are defined according to ๐‘ฃ(๐‘‹)=๐น๐‘ฃ(๐‘‹) and ๐‘€=๐ธ๐‘€๐นโ‰ˆ๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘› with ๐‘ฃ(๐‘€)=๐ธ๐‘ฃ(๐‘€)=๐ธ(๐‘ฃ๐‘‡1(๐‘€),๐‘ฃ๐‘‡2(๐‘€))๐‘‡.

3. Results Concerning Sets of Pair Wise Commuting Matrices

Consider the following sets.

(1)A set of nonzero ๐‘โ‰ฅ2 distinct pair wise commuting matrices ๐€๐ถโˆถ={๐ด๐‘–โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐ด๐‘–,๐ด๐‘—]=0;โˆ€๐‘–,๐‘—โˆˆ๐‘}.(2)The set of matrices MC๐€๐ถโˆถ={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐‘‹,๐ด๐‘–]=0;โˆ€๐ด๐‘–โˆˆ๐€๐ถ} which commute with the set ๐€๐ถ of pair wise commuting matrices.(3)A set of matrices ๐ถ๐€โˆถ={๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐‘‹,๐ด๐‘–]=0;โˆ€๐ด๐‘–โˆˆ๐€} which commute with a given set of nonzero ๐‘ matrices ๐€โˆถ={๐ด๐‘–โˆˆ๐‘๐‘›ร—๐‘›;โˆ€๐‘–โˆˆ๐‘} which are not necessarily pair wise commuting.

The complementary sets of MC๐€๐ถ and ๐ถ๐€ are MC๐€๐ถ and ๐ถ๐€, respectively, so that ๐‘๐‘›ร—๐‘›โˆ‹๐ตโˆˆMC๐€๐ถ if ๐ตโˆ‰MC๐€๐ถ and ๐‘๐‘›ร—๐‘›โˆ‹๐ตโˆˆ๐ถ๐€ if ๐ตโˆ‰๐ถ๐€. Note that ๐ถ๐€๐ถ=MC๐ด๐ถ for a set of pair wise commuting matrices ๐€๐ถ so that the notation MC๐ด๐ถ 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 ๐ด๐‘–โˆˆ๐€๐ถ;โˆ€๐‘–โˆˆ๎€ท๐ด๐‘โŸบ๐‘ฃ๐‘–๎€ธโˆˆ๎™๐‘—(โ‰ ๐‘–)โˆˆ๐‘๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘๎€ท๐ดโŸบ๐‘ฃ๐‘–๎€ธโˆˆ๎™๐‘–+1โ‰ค๐‘—โ‰ค๐‘๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘.(3.1)(ii)Define ๐‘๐‘–๎€ท๐€๐ถ๎€ธ๎๐ดโˆถ=๐‘‡1โŠ•๎€ทโˆ’๐ด1๎€ธ๐ด๐‘‡2โŠ•๎€ทโˆ’๐ด2๎€ธโ‹ฏ๐ด๐‘‡๐‘–โˆ’1โŠ•๎€ทโˆ’๐ด๐‘–โˆ’1๎€ธ๐ด๐‘‡๐‘–+1โŠ•๎€ทโˆ’๐ด๐‘–+1๎€ธโ‹ฏ๐ด๐‘‡๐‘โŠ•๎€ทโˆ’๐ด๐‘๎€ธ๎ž๐‘‡โˆˆ๐‘(๐‘โˆ’1)๐‘›2ร—๐‘›2.(3.2) Then ๐ด๐‘–โˆˆ๐€๐ถ;โˆ€๐‘–โˆˆ๐‘ if and only if ๐‘ฃ(๐ด๐‘–)โˆˆKer๐‘๐‘–(๐€๐ถ);โˆ€๐‘–โˆˆ๐‘.(iii)One has MC๐€๐ถโŽงโŽชโŽจโŽชโŽฉโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎™โˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;๐ด๐‘–โˆˆ๐€๐ถโŽซโŽชโŽฌโŽชโŽญ=๎€ฝ๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎€ท๐€โˆถ๐‘ฃ(๐‘‹)โˆˆKer๐‘๐ถ๎€ธ๎€พโŠƒ๐ถ๐€๐ถโŠƒ๐€๐ถโŠƒ{0}โˆˆ๐‘๐‘›ร—๐‘›,(3.3) where ๐‘(๐€๐ถ)โˆถ=[๐ด๐‘‡1โŠ•(โˆ’๐ด1)๐ด๐‘‡2โŠ•(โˆ’๐ด2)โ‹ฏ๐ด๐‘‡๐‘โŠ•(โˆ’๐ด๐‘)]๐‘‡โˆˆ๐‘๐‘๐‘›2ร—๐‘›2,๐ด๐‘–โˆˆ๐€๐ถ.(iv)One has MC๐€๐ถโŽงโŽชโŽจโŽชโŽฉโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎šโˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดIm๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;๐ด๐‘–โˆˆ๐€๐ถโŽซโŽชโŽฌโŽชโŽญ=๎€ฝ๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎€ท๐€โˆถ๐‘ฃ(๐‘‹)โˆˆIm๐‘๐ถ.๎€ธ๎€พ(3.4)(v)One has ๐ถ๐€โŽงโŽชโŽจโŽชโŽฉโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎™โˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;๐ด๐‘–โŽซโŽชโŽฌโŽชโŽญ=๎€ฝโˆˆ๐€๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎€พ,โˆถ๐‘ฃ(๐‘‹)โˆˆKer๐‘(๐€)(3.5) where ๐‘(๐€)โˆถ=[๐ด๐‘‡1โŠ•(โˆ’๐ด1)๐ด๐‘‡2โŠ•(โˆ’๐ด2)โ‹ฏ๐ด๐‘‡๐‘โŠ•(โˆ’๐ด๐‘)]๐‘‡โˆˆ๐‘๐‘๐‘›2ร—๐‘›2,๐ด๐‘–โˆˆ๐€.(vi)One has ๐ถ๐€โŽงโŽชโŽจโŽชโŽฉโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎šโˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดIm๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;๐ด๐‘–โŽซโŽชโŽฌโŽชโŽญ=๎€ฝโˆˆ๐€๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎€พ.โˆถ๐‘ฃ(๐‘‹)โˆˆIm๐‘(๐€)(3.6)

Concerning Proposition 3.1(v)-(vi), note that if ๐‘‹โˆˆ๐ถ๐€, then ๐‘‹โ‰ 0 since ๐‘๐‘›ร—๐‘›โˆ‹0โˆˆ๐ถ๐€. The following result is related to the rank defectiveness of the matrix ๐‘(๐€๐ถ) and any of their submatrices since ๐€๐ถ is a set of pair wise commuting matrices.

Proposition 3.2. The following properties hold: ๐‘›2๎€ท๐€>rank๐‘๐ถ๎€ธโ‰ฅrank๐‘๐‘–๎€ท๐€๐ถ๎€ธ๎€ท๐ดโ‰ฅrank๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐ด๐‘—โˆˆ๐€๐ถ;โˆ€๐‘–,๐‘—โˆˆ๐‘(3.7) and, equivalently, ๎€ท๐‘det๐‘‡๎€ท๐€๐ถ๎€ธ๐‘๎€ท๐€๐ถ๎€ท๐‘๎€ธ๎€ธ=det๐‘‡๐‘–๎€ท๐€๐ถ๎€ธ๐‘๐‘–๎€ท๐€๐ถ๎€ท๐ด๎€ธ๎€ธ=det๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ=0;โˆ€๐ด๐‘—โˆˆ๐€๐ถ;โˆ€๐‘–,๐‘—โˆˆ๐‘›.(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 ๐‘(๐€) of some set ๐€ of ๐‘ square real ๐‘›-matrices is discussed. Such a condition may be useful as a practical test to elucidate the existence of a nonzero ๐‘›-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 ๐‘›-square real matrices ๐€โˆถ={๐ด1,๐ด2,โ€ฆ,๐ด๐‘} for any integer ๐‘โ‰ฅ1 and define matrices: ๐‘๐‘–(๎€บ๐ด๐€)โˆถ=๐‘‡1โŠ•๎€ทโˆ’๐ด1๎€ธ๐ด๐‘‡2โŠ•๎€ทโˆ’๐ด2๎€ธโ‹ฏ๐ด๐‘‡๐‘–โˆ’1โŠ•๎€ทโˆ’๐ด๐‘–โˆ’1๎€ธ๐ด๐‘‡๐‘–+1โŠ•๎€ทโˆ’๐ด๐‘–+1๎€ธโ‹ฏ๐ด๐‘‡๐‘โŠ•๎€ทโˆ’๐ด๐‘๎€ธ๎€ป๐‘‡,๎๐ด๐‘(๐€)โˆถ=๐‘‡1โŠ•๎€ทโˆ’๐ด1๎€ธ๐ด๐‘‡2โŠ•๎€ทโˆ’๐ด2๎€ธโ‹ฏ๐ด๐‘‡๐‘โŠ•๎€ทโˆ’๐ด๐‘๎€ธ๎ž๐‘‡.(3.9) Then, the following properties hold: (i)rank(๐ด๐‘–โŠ•(โˆ’๐ด๐‘–))โ‰คrank๐‘๐‘–(๐€)โ‰คrank๐‘(๐€)<๐‘›2;โˆ€๐‘–โˆˆ๐‘.(ii)โ‹‚๐‘–โˆˆ๐‘Ker(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–))โ‰ {0} so that โˆƒ๐‘‹(โ‰ 0)โˆˆ๐ถ๐€,๐‘‹โˆˆ๐ถ๐€๎™โŸบ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–,๎€ธ๎€ธ๐‘‹โˆˆ๐ถ๐€๎šโŸบ๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘๎€ท๐ดIm๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–.๎€ธ๎€ธ(3.10)(iii)If ๐€=๐€๐ถ is a set of pair wise commuting matrices then ๐‘ฃ๎€ท๐ด๐‘–๎€ธโˆˆ๎™๐‘—โˆˆ๐‘โงต๐‘–๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘๎€ท๐ดโŸบ๐‘ฃ๐‘–๎€ธโˆˆ๎™๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘๎€ท๐ดโŸบ๐‘ฃ๐‘–๎€ธโˆˆ๎™๐‘–โˆˆ๐‘โงต{๐‘–}๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘.(3.11)(iv)One has M๐€๐ถโŽงโŽชโŽจโŽชโŽฉโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›๎™โˆถ๐‘ฃ(๐‘‹)๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ธ๎€ธ,โˆ€๐ด๐‘–โˆˆ๐€๐ถโŽซโŽชโŽฌโŽชโŽญโŠƒ๐€๐ถโˆช{0}โˆˆ๐‘๐‘›ร—๐‘›(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 ๐ถ๐€โงต{0}โ‰ โˆ… because all nonzero ๐‘๐‘›ร—๐‘›โˆ‹ฮ›=diag(๐œ†๐œ†โ‹ฏ๐œ†)โˆˆ๐ถ๐€ for any ๐‘โˆ‹๐œ†โ‰ 0 and any set of matrices ๐€. Note that Theorem 3.3(iii) establishes that ๐‘ฃ(๐ด๐‘–โ‹‚)โˆˆ๐‘–โˆˆ๐‘โงต{๐‘–}Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—));โˆ€๐‘–โˆˆ๐‘ is a necessary and sufficient condition for the set to be a set of commuting matrices ๐€ being simpler to test (by taking advantage of the symmetry property of the commutators) than the equivalent condition ๐‘ฃ(๐ด๐‘–โ‹‚)โˆˆ๐‘–โˆˆ๐‘Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—));โˆ€๐‘–โˆˆ๐‘. 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 ๐‘›-square real matrices ๐€={๐ด1,๐ด2,โ€ฆ,๐ด๐‘}.
(i) The set ๐ถ๐€ of matrices ๐‘‹โˆˆ๐‘๐‘›ร—๐‘› which commute with all matrices in ๐€ is defined by: ๐ถ๐€๎ƒฏโˆถ=๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘๎™๐‘–=1๎‚€๐ฝKer๎‚ƒ๎‚€๐ด๐‘–โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘–๎€ท๐‘ƒ๎‚๎‚๐‘–โˆ’1โŠ—๐‘ƒ๐‘–โˆ’๐‘‡๎€ธ๎ƒฐ=๎ƒฏ๎‚„๎‚๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘๎™๐‘–=1๎€ท๐‘ƒIm๎€ท๎€ท๐‘–โŠ—๐‘ƒ๐‘–โˆ’1๐‘Œ๎€ธ๎€ท๐‘–๎€ธ๎€ธ๎€ธโˆง๐‘Œ๐‘–๎‚€๐ฝโˆˆKer๐ด๐‘–โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘–๎‚๎‚;โˆ€๐‘–โˆˆ๐‘๎ƒฐ=๎ƒฏ๐‘‹โˆˆ๐‘๐‘›ร—๐‘›โˆถ๐‘ฃ(๐‘‹)โˆˆ๐‘๎™๐‘–=1๎€ท๐‘ƒIm๎€ท๎€ท๐‘–โŠ—๐‘ƒ๐‘–โˆ’1๎€ธ(๐‘Œ)๎€ธ๎€ธ,๐‘Œโˆˆ๐‘๎™๐‘–=1๎‚€๎‚€๐ฝKer๐ด๐‘–โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘–๎ƒฐ,๎‚๎‚๎‚(3.13) where ๐‘ƒ๐‘–โˆˆ๐‘๐‘›ร—๐‘› is a nonsingular transformation matrix such that ๐ด๐‘–โˆผ๐ฝ๐ด๐‘–=๐‘ƒ๐‘–โˆ’1๐ด๐‘–๐‘ƒ๐‘–, ๐ฝ๐ด๐‘– being the Jordan canonical form of ๐ด๐‘–.
(ii) One has ๎€ฝdimspan๐‘ฃ(๐‘‹)โˆถ๐‘‹โˆˆ๐ถ๐€๎€พโ‰คmin๐‘–โˆˆ๐‘๎‚€๎‚€๐ฝdimKer๐ด๐‘–โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘–๎‚๎‚๎‚=min๐‘–โˆˆ๐‘๐œˆ๐‘–(0)=min๐‘–โˆˆ๐‘๎ƒฉ๐œŒ๐‘–๎“๐‘—=1๐œˆ2๐‘–๐‘—๎ƒชโ‰คmin๐‘–โˆˆ๐‘๎ƒฉ๐œŒ๐‘–๎“๐‘–=1๐œ‡2๐‘–๐‘—๎ƒชโ‰คmin๐‘–โˆˆ๐‘๎€ท๐œ‡๐‘–๎€ธ,(0)(3.14) where ๐œˆ๐‘–(0) and ๐œˆ๐‘–๐‘— are, respectively, the geometric multiplicities of 0โˆˆ๐œŽ(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)) and ๐œ†๐‘–๐‘—โˆˆ๐œŽ(๐ด๐‘–) and ๐œ‡๐‘–(0) and ๐œ‡๐‘–๐‘— are, respectively, the algebraic multiplicities of 0โˆˆ๐œŽ(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)) and ๐œ†๐‘–๐‘—โˆˆ๐œŽ(๐ด๐‘–); โˆ€๐‘—โˆˆ๐œŒ๐‘– (the number of distinct eigenvalues of ๐ด๐‘–), โˆ€๐‘–โˆˆ๐‘.
(iii) The set ๐€ consists of pair wise commuting matrices, namely ๐ถ๐€=MC๐€, if and only if ๐‘ฃ(๐ด๐‘—โ‹‚)โˆˆ๐‘๐‘–(โ‰ ๐‘—)=1(Ker[(๐ฝ๐ด๐‘–โŠ•(โˆ’๐ฝ๐‘‡๐ด๐‘–))(๐‘ƒ๐‘–โˆ’1โŠ—๐‘ƒ๐‘–โˆ’๐‘‡)]); โˆ€๐‘—โˆˆ๐‘. Equivalent conditions follow from the second and third equivalent definitions of ๐ถ๐€ in Property (i).

Theorems 3.3 and 3.4 are concerned with MC๐€โ‰ {0}โˆˆ๐‘๐‘›ร—๐‘› 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 ๐ด=๐ดre+๐ข๐ดim and ๐ต=๐ตre+๐ข๐ตim be complex matrices in ๐‚๐‘›ร—๐‘› with ๐ดre and ๐ตre being their respective real parts, and ๐ดim and ๐ตim, all in ๐‘๐‘›ร—๐‘›, their respective imaginary parts, and โˆš๐ข=โˆ’1 is the imaginary complex unity. Direct computations with the commutator of ๐ด and ๐ต yield []=๐ด๐ด,๐ต๎€ท๎€บre,๐ตre๎€ปโˆ’๎€บ๐ดim,๐ตim๐ด๎€ป๎€ธ+๐ข๎€ท๎€บim,๐ตre๎€ป+๎€บ๐ดre,๐ตim๎€ป๎€ธ.(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 ๐ตโˆˆ๐ถ๐ดโ‡”(([๐ดre,๐ตre]=[๐ดim,๐ตimโ‹€])([๐ดim,๐ตre]=[๐ตim,๐ดre])).

Proposition 4.2. One has (๐ตreโˆˆ(๐ถ๐ดreโˆฉ๐ถ๐ดim)โ‹€๐ตimโˆˆ(๐ถ๐ดimโˆฉ๐ถ๐ดre))โ‡’๐ตโˆˆ๐ถ๐ด.

Proposition 4.3. One has (๐ดreโˆˆ(๐ถ๐ตreโˆฉ๐ถ๐ตim)โ‹€๐ดimโˆˆ(๐ถ๐ตimโˆฉ๐ถ๐ตre))โ‡’๐ตโˆˆ๐ถ๐ด.

Proposition 4.1 yields to the subsequent result.

Theorem 4.4. The following properties hold.
(i) Assume that the matrices ๐ด and ๐ตre are given. Then, ๐ตโˆˆ๐ถ๐ด if and only if ๐ตim satisfies the following linear algebraic equation: ๎ƒฌ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎ƒญ๐‘ฃ๎€ท๐ตre๎€ธ=๎ƒฌ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎ƒญ๐‘ฃ๎€ท๐ตim๎€ธ(4.2) for which a necessary condition is ๎ƒฌ๐ดrankimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎ƒญ๎ƒฌ๐ด=rankimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎ƒฉ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎ƒช๐‘ฃ๎€ท๐ตre๎€ธ๎ƒญ.(4.3)
(ii) Assume that the matrices ๐ด and ๐ตim๐‘’ are given. Then, ๐ตโˆˆ๐ถ๐ด if and only if ๐ตre satisfies (4.2) for which a necessary condition is ๎ƒฌ๐ดrankreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎ƒญ๎ƒฌ๐ด=rankreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎ƒฉ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎ƒช๐‘ฃ๎€ท๐ตim๎€ธ๎ƒญ.(4.4)
(iii) Also, โˆƒ๐ตโ‰ 0 such that ๐ตโˆˆ๐ถ๐ด with ๐ตre=0 and โˆƒ๐ตโ‰ 0 such that ๐ตโˆˆ๐ถ๐ด with ๐ตim=0.

A more general result than Theorem 4.4 is the following.

Theorem 4.5. The following properties hold.
(i) ๐ตโˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘› if and only if ๐‘ฃ(๐ต) is a solution to the following linear algebraic system: ๎ƒฌ๐ดreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎€ทโˆ’๐ดim๎€ธโŠ•๎€ท๐ด๐‘‡im๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎€ทโˆ’๐ดre๎€ธโŠ•๎€ท๐ด๐‘‡re๎€ธ๐‘ฃ๎€ท๐ต๎ƒญ๎ƒฌre๎€ธ๐‘ฃ๎€ท๐ตim๎€ธ๎ƒญ=0.(4.5) Nonzero solutions ๐ตโˆˆ๐ถ๐ด, satisfying ๎ƒฌ๐‘ฃ๎€ท๐ตre๎€ธ๐‘ฃ๎€ท๐ตim๎€ธ๎ƒญ๎ƒฌ๐ดโˆˆKerreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎€ทโˆ’๐ดim๎€ธโŠ•๎€ท๐ด๐‘‡im๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎€ทโˆ’๐ดre๎€ธโŠ•๎€ท๐ด๐‘‡re๎€ธ๎ƒญ,(4.6) always exist since ๎ƒฌ๐ดKerreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎€ทโˆ’๐ดim๎€ธโŠ•๎€ท๐ด๐‘‡im๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎€ทโˆ’๐ดre๎€ธโŠ•๎€ท๐ด๐‘‡re๎€ธ๎ƒญโ‰ {0}โˆˆ๐‘2๐‘›2,(4.7) and equivalently, since ๎ƒฌ๐ดrankreโŠ•๎€ทโˆ’๐ด๐‘‡re๎€ธ๎€ทโˆ’๐ดim๎€ธโŠ•๎€ท๐ด๐‘‡im๎€ธ๐ดimโŠ•๎€ทโˆ’๐ด๐‘‡im๎€ธ๎€ทโˆ’๐ดre๎€ธโŠ•๎€ท๐ด๐‘‡re๎€ธ๎ƒญ<2๐‘›2.(4.8)
(ii) Property (ii) is equivalent to ๐ตโˆˆ๐ถ๐ดโŸบ๎€ท๎€ท๐ดโŠ•โˆ’๐ดโˆ—๐‘ฃ๎€ธ๎€ธ(๐ต)=0(4.9) which has always nonzero solutions since rank(๐ดโŠ•(โˆ’๐ดโˆ—))<๐‘›2.

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 ๎ƒฌ๐ด๐‘—reโŠ•๎€ทโˆ’๐ด๐‘‡2re๎€ธ๎€ทโˆ’๐ด๐‘—im๎€ธโŠ•๎‚€๐ด๐‘‡๐‘—im๎‚๐ด๐‘—imโŠ•๎€ทโˆ’๐ด๐‘‡2im๎€ธ๎€ทโˆ’๐ด๐‘—2re๎€ธโŠ•๎€ท๐ด๐‘‡๐‘—re๎€ธ๎ƒญ(4.10) for each ๐‘—โˆˆ๐‘ 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 ๐€โˆถ={๐ด๐‘–โˆˆ๐‚๐‘›ร—๐‘›โˆถ๐‘–โˆˆ๐‘} and ๐ถ๐€โˆถ={๐‘‹โˆˆ๐‚๐‘›ร—๐‘›โˆถ[๐‘‹,๐ด๐‘–]=0;๐ด๐‘–โˆˆ๐€,โˆ€๐‘–โˆˆ๐‘} for ๐‘โ‰ฅ2. Thus, ๐ถ๐€โˆ‹๐‘‹=๐‘‹re+๐ข๐‘‹re if and only if โŽกโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽขโŽฃ๐ด1reโŠ•๎€ทโˆ’๐ด๐‘‡1re๎€ธ๎€ทโˆ’๐ด1im๎€ธโŠ•๎€ท๐ด๐‘‡1im๎€ธ๐ด1imโŠ•๎€ทโˆ’๐ด๐‘‡1im๎€ธ๎€ทโˆ’๐ด1re๎€ธโŠ•๎€ท๐ด๐‘‡1re๎€ธ๐ด2reโŠ•๎€ทโˆ’๐ด๐‘‡2re๎€ธ๎€ทโˆ’๐ด2im๎€ธโŠ•๎€ท๐ด๐‘‡2im๎€ธ๐ด2imโŠ•๎€ทโˆ’๐ด๐‘‡2im๎€ธ๎€ทโˆ’๐ด2re๎€ธโŠ•๎€ท๐ด๐‘‡2re๎€ธโ‹ฎ๐ด๐‘reโŠ•๎€ทโˆ’๐ด๐‘‡๐‘re๎€ธ๎€ทโˆ’๐ด๐‘im๎€ธโŠ•๎‚€๐ด๐‘‡๐‘im๎‚๐ด๐‘imโŠ•๎‚€โˆ’๐ด๐‘‡๐‘im๎‚๎€ทโˆ’๐ด๐‘re๎€ธโŠ•๎€ท๐ด๐‘‡๐‘re๎€ธโŽคโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฅโŽฆ๎ƒฌ๐‘ฃ๎€ท๐‘‹re๎€ธ๐‘ฃ๎€ท๐‘‹im๎€ธ๎ƒญ=0,(4.11) and a nonzero solution ๐‘‹โˆˆ๐ถ๐€ exists since the rank of the coefficient matrix of (4.11) is less than 2๐‘›2.
(ii) Consider the sets of nonzero distinct commuting complex matrices ๐€๐ถโˆถ={๐ด๐‘–โˆˆ๐‚๐‘›ร—๐‘›โˆถ๐‘–โˆˆ๐‘} and MC๐€โˆถ={๐‘‹โˆˆ๐‚๐‘›ร—๐‘›โˆถ[๐‘‹,๐ด๐‘–]=0;๐ด๐‘–โˆˆ๐€,โˆ€๐‘–โˆˆ๐‘} for ๐‘โ‰ฅ2. Thus, MC๐€โˆ‹๐‘‹=๐‘‹re+๐ข๐‘‹re if and only if ๐‘ฃ(๐‘‹r๐‘’) and ๐‘ฃ(๐‘‹im) are solutions to (4.11).
(iii) Properties (i) and (ii) are equivalently formulated by from the algebraic set of complex equations: ๎€บ๐ดโˆ—1โŠ•๎€ทโˆ’๐ด1๎€ธ๐ดโˆ—2โŠ•๎€ทโˆ’๐ด2๎€ธโ‹ฏ๐ดโˆ—๐‘โŠ•๎€ทโˆ’๐ด๐‘๎€ธ๎€ปโˆ—๐‘ฃ(๐‘‹)=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 ๐‘“โˆถ๐‚โ†’๐‚ be an analytic function in an open set ๐ทโŠƒ๐œŽ(๐ด) for some matrix ๐ดโˆˆ๐‚๐‘›ร—๐‘› and let ๐‘(๐œ†) be a polynomial fulfilling ๐‘(๐‘–)(๐œ†๐‘˜)=๐‘“(๐‘–)(๐œ†๐‘˜); โˆ€๐‘˜โˆˆ๐œŽ(๐ด), โˆ€๐‘–โˆˆ๐‘š๐‘˜โˆ’1โˆช{0}; โˆ€๐‘˜โˆˆ๐œ‡ (the number of distinct elements in ๐œŽ(๐ด)), where ๐‘š๐‘˜ is the index of ๐œ†๐‘˜, that is, its multiplicity in the minimal polynomial of ๐ด. Then, ๐‘“(๐ด) is a function of a matrix ๐ด if ๐‘“(๐ด)=๐‘(๐ด), [8]. Some results follow concerning the commutators of functions of matrices.

Theorem 4.8. Consider a nonzero matrix ๐ตโˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘› for any given nonzero ๐ดโˆˆ๐‚๐‘›ร—๐‘›. Then, ๐‘“(๐ต)โˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘›, and equivalently ๐‘ฃ(๐‘“(๐ต))โˆˆKer(๐ดโŠ•(โˆ’๐ดโˆ—)), for any function ๐‘“โˆถ๐‚๐‘›ร—๐‘›โ†’๐‚๐‘›ร—๐‘› of the matrix ๐ต.

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

(1)๐ดโˆˆ๐ถ๐ด;โˆ€๐ดโˆˆ๐‚๐‘›ร—๐‘›,(2)One has [][][]=[]=๐ด,๐ต=0โŸน๐ด,๐‘”(๐ต)=0โŸน๐‘“(๐ด),๐‘”(๐ต)๐‘(๐ด),๐‘”(๐ต)๐œ‡๎“๐‘–=0๐›ผ๐‘–๎€บ๐ด๐‘–๎€ป=,๐‘”(๐ต)๐œ‡๎“๐‘–=0๐›ผ๐‘–๐ด๐‘–โˆ’1[]๐ด,๐‘”(๐ต)=0โŸบ๐‘”(๐ต)โˆˆ๐ถ๐‘“(๐ด)โˆฉ๐‚๐‘›ร—๐‘›,(4.13) where ๐‘“(๐ด)=๐‘(๐ด), from the definition of ๐‘“ being a function of the matrix ๐ด, with ๐‘(๐œ†) being a polynomial fulfilling ๐‘(๐‘–)(๐œ†๐‘˜)=๐‘“(๐‘–)(๐œ†๐‘˜); โˆ€๐‘˜โˆˆ๐œŽ(๐ด), โˆ€๐‘–โˆˆ๐‘š๐‘˜โˆ’1โˆช{0}; โˆ€๐‘˜โˆˆ๐œ‡ (the number of distinct elements in ๐œŽ(๐ด)), where ๐‘š๐‘˜ is the index of ๐œ†๐‘˜, that is, its multiplicity in the minimal polynomial of ๐ด.(3)Theorem 4.8 is extendable for any countable set {๐‘“๐‘–(๐ต)} of matrix functions of ๐ต.

Corollary 4.9. Consider a nonzero matrix ๐ตโˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘› for any given nonzero ๐ดโˆˆ๐‚๐‘›ร—๐‘›. Then, ๐‘”(๐ต)โˆˆ๐ถ๐‘“(๐ด)โˆฉ๐‚๐‘›ร—๐‘› for any function ๐‘“โˆถ๐‚๐‘›ร—๐‘›โ†’๐‚๐‘›ร—๐‘› of the matrix ๐ด and any function ๐‘”โˆถ๐‚๐‘›ร—๐‘›โ†’๐‚๐‘›ร—๐‘› of the matrix ๐ต.

Corollary 4.10. ๐‘“(๐ด)โˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘›, and equivalently ๐‘ฃ(๐‘“(๐ด))โˆˆKer(๐ดโŠ•(โˆ’๐ดโˆ—)), for any function ๐‘“โˆถ๐‚๐‘›ร—๐‘›โ†’๐‚๐‘›ร—๐‘› of the matrix ๐ด.

Corollary 4.11. If ๐ตโˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘› then any countable set of function matrices {๐‘“๐‘–(๐ต)} is ๐ถ๐ด and in MC๐ด.

Corollary 4.12. Consider any countable set of function matrices ๐ถ๐นโˆถ={๐‘“๐‘–(๐ด);โˆ€๐‘–โˆˆ๐‘}โŠ‚๐ถ๐ด for any given nonzero ๐ดโˆˆ๐‚๐‘›ร—๐‘›. Then, โ‹‚๐‘“๐‘–โˆˆ๐ถ๐น(Ker(๐‘“๐‘–(๐ด)โŠ•(โˆ’๐‘“๐‘–(๐ดโˆ—))))โŠƒKer(๐ดโŠ•(โˆ’๐ดโˆ—)).

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 ๐ตโˆˆ๐ถ๐ดโˆฉ๐‚๐‘›ร—๐‘›. Thus, ฮ›๐ตโˆˆ๐ถฮ›๐ดโˆฉ๐‚๐‘›ร—๐‘› provided that there exists a nonsingular matrix ๐‘‡โˆˆ๐‚๐‘›ร—๐‘› such that ฮ›๐ด=๐‘‡โˆ’1๐ด๐‘‡ and ฮ›๐ต=๐‘‡โˆ’1๐ต๐‘‡.

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.


A. Proofs of the Results of Section 2

Proof of Proposition 2.1. (i)-(ii) First note by inspection that โˆ…โ‰ ๐ถ๐ดโŠƒ{0,๐ด}; โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›. Also, []๎€ท๐ด,๐‘‹=๐ด๐‘‹โˆ’๐‘‹๐ด=๐ดโŠ—๐ผ๐‘›โˆ’๐ผ๐‘›โŠ—๐ด๐‘‡๎€ธ๐‘ฃ=๎€ท๎€ท(๐‘‹)๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธ๐‘ฃ(๐‘‹)=0โŸน๐‘ฃ(๐‘‹)โˆˆKer๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ(A.1) and Proposition 2.1(i)-(ii) has been proved since there is an isomorphism ๐‘“โˆถ๐‘๐‘›2โ†”๐‘๐‘›ร—๐‘› defined by ๐‘“(๐‘ฃ(๐‘‹))=๐‘‹; โˆ€๐‘‹โˆˆ๐‘๐‘›ร—๐‘› for ๐‘ฃ(๐‘‹)=(๐‘ฅ๐‘‡1,๐‘ฅ๐‘‡2,โ€ฆ,๐‘ฅ๐‘‡๐‘›)๐‘‡โˆˆ๐‘๐‘›2 if ๐‘ฅ๐‘‡๐‘–โˆถ=(๐‘ฅ๐‘–1,๐‘ฅ๐‘–2,โ€ฆ,๐‘ฅ๐‘–๐‘›) is the ith row of the square matrix ๐‘‹.
(iii) It is a direct consequence of Proposition 2.1(iii) and the symmetry property of the commutator of two commuting matrices ๐ตโˆˆ๐ถ๐ดโ‡”[๐ด,๐ต]=[๐ต,๐ด]=0โ‡”๐ดโˆˆ๐ถ๐ต.

Proof of Proposition 2.2. [๐ด,๐ด]=0;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›โ‡’โˆƒ๐‘๐‘›2โˆ‹0โ‰ ๐‘ฃ(๐ด)โˆˆKer(๐ดโŠ•(โˆ’๐ด๐‘‡)); โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›. As a result, ๎€ท๎€ทKer๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธโ‰ 0โˆˆ๐‘๐‘›2;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›๎€ท๎€ทโŸบrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ<๐‘›2;โˆ€๐ดโˆˆ๐‘๐‘›ร—๐‘›(A.2) so that 0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)).
Also, โˆƒ๐‘‹(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘›โˆถ[๐ด,๐‘‹]=0โ‡”๐‘‹โˆˆ๐ถ๐ด since Ker(๐ดโŠ•(โˆ’๐ด๐‘‡))โ‰ 0โˆˆ๐‘๐‘›2.
Proposition 2.2 has been proved.

Proof of Theorem 2.3. (i) Note that ๐œŽ๎€ท๐ด(๐ด)=๐œŽ๐‘‡๎€ธ๎€ท๎€ทโŸน๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€ฝ๎€ธ๎€ธโˆถ=๐‚โˆ‹๐œ‚=๐œ†๐‘˜โˆ’๐œ†โ„“;โˆ€๐œ†๐‘˜,๐œ†โ„“โˆˆ๐œŽ(๐ด);โˆ€๐‘˜,โ„“โˆˆ๐‘›๎€พ=๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡โˆช๎€ธ๎€ธ๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡,๎€ธ๎€ธ(A.3) where ๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡=๎€ฝ๎€ท๎€ท๎€ธ๎€ธ๐œ†โˆˆ๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€พ,๎€ธ๎€ธโˆถ๐œ†=0๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡=๎€ฝ๎€ท๎€ท๎€ธ๎€ธ๐œ†โˆˆ๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€พ๎€ท๎€ท๎€ธ๎€ธโˆถ๐œ†โ‰ 0=๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธโงต๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡.๎€ธ๎€ธ(A.4) Furthermore, ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡))โˆถ={๐‚โˆ‹๐œ†=๐œ†๐‘—โˆ’๐œ†๐‘–โˆถ๐œ†๐‘–,๐œ†๐‘—โˆˆ๐œŽ(๐ด);โˆ€๐‘–,๐‘—โˆˆ๐‘›} and ๐‘ง๐‘–โŠ—๐‘ฅ๐‘— is a right eigenvector of ๐ดโŠ•(โˆ’๐ด๐‘‡) associated with its eigenvalue ๐œ†๐‘—๐‘–=๐œ†๐‘—โˆ’๐œ†๐‘–. ๐œ†=๐œ†๐‘—โˆ’๐œ†๐‘–โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) has algebraic and geometric multiplicities ๐œ‡๐‘—๐‘– and ๐œˆ๐‘—๐‘–, respectively; โˆ€๐‘–,๐‘—โˆˆ๐‘›, since ๐‘ฅ๐‘— and ๐‘ง๐‘– are, respectively, the right eigenvectors of ๐ด and ๐ด๐‘‡ with associated eigenvalues ๐œ†๐‘— and ๐œ†๐‘–;โˆ€๐‘–,๐‘—โˆˆ๐‘›.
Let ๐ฝ๐ด be the Jordan canonical form of ๐ด. It is first proved that there exists a nonsingular ๐‘‡โˆˆ๐‘๐‘›2ร—๐‘›2 such that ๐ฝ๐ดโŠ•(โˆ’๐ฝ๐ด๐‘‡)=๐‘‡โˆ’1(๐ดโŠ•(โˆ’๐ด๐‘‡))๐‘‡. The proof is made by direct verification by using the properties of the Kronecker product, with ๐‘‡=๐‘ƒโŠ—๐‘ƒ๐‘‡ for a nonsingular ๐‘ƒโˆˆ๐‘๐‘›ร—๐‘› such that ๐ดโˆผ๐ฝ๐ด=๐‘ƒโˆ’1๐ด๐‘ƒ, as follows: ๐‘‡โˆ’1๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ธ๎€ธ๐‘‡=๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธโˆ’1๎€ท๐ดโŠ—๐ผ๐‘›๎€ธ๎€ท๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธโˆ’๎€ท๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธโˆ’1๎€ท๐ผ๐‘›โŠ—๐ด๐‘‡๎€ธ๎€ท๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธ=๎€ท๐‘ƒโˆ’1๎€ธโŠ—๎€ท๐‘ƒ๐ด๐‘ƒโˆ’๐‘‡๐ผ๐‘›๐‘ƒ๐‘‡๎€ธโˆ’๎€ท๐‘ƒโˆ’1๐ผ๐‘›๐‘ƒ๎€ธโŠ—๎€ท๐‘ƒโˆ’๐‘‡๐ด๐‘‡๐‘ƒ๐‘‡๎€ธ=๎€ท๐‘ƒโˆ’1๎€ธ๐ด๐‘ƒโŠ—๐ผ๐‘›โˆ’๐ผ๐‘›โŠ—๎€ท๐‘ƒโˆ’๐‘‡๐ด๐‘‡๐‘ƒ๐‘‡๎€ธ=๐ฝ๐ดโŠ—๐ผ๐‘›โˆ’๐ผ๐‘›โŠ—๐ฝ๐ด๐‘‡=๐ฝ๐ดโŠ—๐ผ๐‘›+๐ผ๐‘›โŠ—๎€ทโˆ’๐ฝ๐ด๐‘‡๎€ธ=๐ฝ๐ดโŠ•๎€ทโˆ’๐ฝ๐ด๐‘‡๎€ธ(A.5) and the result has been proved. Thus, rank(๐ดโŠ•(โˆ’๐ด๐‘‡))=rank(๐ฝ๐ดโŠ•(โˆ’๐ฝ๐ด๐‘‡)). It turns out that ๐‘ƒ is, furthermore, unique except for multiplication by any nonzero real constant. Otherwise, if ๐‘‡โ‰ ๐‘ƒโŠ—๐‘ƒ๐‘‡, then there would exist a nonsingular ๐‘„โˆˆ๐‘๐‘›ร—๐‘› with ๐‘„โ‰ ๐›ผ๐ผ๐‘›;โˆ€๐›ผโˆˆ๐‘ such that ๐‘‡=๐‘„(๐‘ƒโŠ—๐‘ƒ๐‘‡)โˆ’1๐‘„ so that ๐‘‡โˆ’1(๐ดโŠ•(โˆ’๐ด๐‘‡))๐‘‡โ‰ ๐ฝ๐ดโŠ•(โˆ’๐ฝ๐ด๐‘‡) provided that ๎€ท๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธโˆ’1๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ๎€ท๐‘ƒโŠ—๐‘ƒ๐‘‡๎€ธ=๐ฝ๐ดโŠ•๎€ทโˆ’๐ฝ๐ด๐‘‡๎€ธ.(A.6) Thus, note that ๎€ท๎€ทcard๐œŽ๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ=๐‘›2=๐œ‡๎“๐‘–=1๐œ‡๐‘–๐‘–=๎ƒฉ๐œ‡๎“๐‘–=1๐œ‡๐‘–๎ƒช2โ‰ฅ๐œ‡(0)=๐œ‡๎“๐‘–=1๐œ‡๐‘–๐‘–=๐œ‡๎“๐‘–=1๐œ‡2๐‘–โ‰ฅ๐œˆโ‰ฅ๐œˆ(0)=๐œ‡๎“๐‘–=1๐œˆ๐‘–๐‘–=๐œ‡๎“๐‘–=1๐œˆ2๐‘–=๎ƒฉ๐œ‡๎“๐œ‡๐‘–=1๎“๐‘—=1๐œˆ๐‘–๐‘—๎ƒช2โˆ’2๐œ‡๎“๐‘–=10๐‘ฅ0200๐‘‘๐œ‡๎“๐‘—(โ‰ ๐‘–)=1๐œˆ๐‘–๐‘—=๐œˆโˆ’2๐œ‡๎“๐‘–=10๐‘ฅ0200๐‘‘๐œ‡๎“๐‘—(โ‰ ๐‘–)=1๐œˆ๐‘–๐‘—โ‰ฅ๐‘›.(A.7) Those results follow directly from the properties of the Kronecker sum ๐ดโŠ•๐ต of n-square real matrices ๐ด and ๐ต=โˆ’๐ด๐‘‡ since direct inspection leads to the following.(1)0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) with algebraic multiplicity โˆ‘๐œ‡(0)โ‰ฅ๐œ‡๐‘–=1๐œ‡๐‘–๐‘–=โˆ‘๐œ‡๐‘–=1๐œ‡2๐‘–โ‰ฅโˆ‘๐œ‡๐‘–=1๐œˆ2๐‘–โ‰ฅ๐‘› since there are at least โˆ‘๐‘›๐‘–=1๐œ‡2๐‘– zeros in ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) (i.e., the algebraic multiplicity of 0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) is at least โˆ‘๐‘›๐‘–=1๐œ‡2๐‘–) and since ๐œˆ๐‘–โ‰ฅ1; โˆ€๐‘–โˆˆ๐‘›. Also, a simple computation of the number of eigenvalues of ๐ดโŠ•(โˆ’๐ด๐‘‡) yields card๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡))=๐‘›2=โˆ‘๐œ‡๐‘–=1๐œ‡๐‘–๐‘–โˆ‘=(๐œ‡๐‘–=1๐œ‡๐‘–)2.(2)The number of linearly independent vectors in ๐‘† is โˆ‘๐œˆ=๐œ‡๐‘–=1โˆ‘๐œ‡๐‘—=1๐œˆ๐‘–๐‘—โˆ‘=(๐œ‡๐‘–=1๐œˆ๐‘–)2โ‰ฅโˆ‘๐œ‡๐‘–=1๐œˆ๐‘–๐‘–=โˆ‘๐œ‡๐‘–=1๐œˆ2๐‘– since the total number of Jordan blocks in the Jordan canonical form of ๐ด is โˆ‘๐œ‡๐‘–=1๐œˆ๐‘–.(3)The number of Jordan blocks associated with 0โˆˆ๐œŽ(๐ดโŠ•(โˆ’๐ด๐‘‡)) in the Jordan canonical form of (๐ดโŠ•(โˆ’๐ด๐‘‡)) is โˆ‘๐œˆ(0)=๐œ‡๐‘–=1๐‘ฃ2๐‘–โ‰ค๐œˆ, with ๐œˆ๐‘–๐‘–=๐œˆ2๐‘–๐‘–; โˆ€๐‘–โˆˆ๐‘›. Thus, card๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡=๎€ธ๎€ธ๐œ‡๎“๐‘–=1๐œ‡๐‘–๐‘–=๐œ‡๎“๐‘–=1๐œ‡2๐‘–,card๐œŽ0๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ=๐‘›2โˆ’๐œ‡๎“๐‘–=1๐œ‡2๐‘–,๎€ท๎€ทrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ=๐‘›2โˆ’๐œˆ(0)=๐‘›2โˆ’๐œ‡๎“๐‘–=1๐‘ฃ2๐‘–๎€ท๎€ท,dimKer๐ดโŠ•โˆ’๐ด๐‘‡=๎€ธ๎€ธ๐œˆ(0)=๐œ‡๎“๐‘–=1๐‘ฃ2๐‘–.(A.8)(4)There are at least ๐œˆ(0) linearly independent vectors in ๐‘†โˆถ=span{๐‘ง๐‘–โŠ—๐‘ฅ๐‘—,โˆ€๐‘–,๐‘—โˆˆ๐‘›}. Also, the total number of Jordan blocks in the Jordan canonical form of (๐ดโŠ•(โˆ’๐ด๐‘‡)) is โˆ‘๐œˆ=dim๐‘†=(๐œ‡๐‘–=1โˆ‘๐œ‡๐‘—=1๐œˆ๐‘–๐‘—โˆ‘)=(๐œ‡๐‘–=1๐œˆ๐‘–)2=โˆ‘๐œˆ(0)+2๐œ‡๐‘–=1โˆ‘๐œ‡๐‘—(โ‰ ๐‘–)=1๐œˆ๐‘–๐‘—โ‰ฅ๐œˆ(0).
Property (i) has been proved. Property (ii) follows directly from the orthogonality in ๐‘๐‘›2of its range and null subspaces.

Proof of Theorem 2.4. First note from Proposition 2.1 that ๐‘‹โˆˆ๐ถ๐ด if and only if (๐ดโŠ•(โˆ’๐ด๐‘‡))๐‘ฃ(๐‘‹)=0 since ๐‘ฃ(๐‘‹)โˆˆKer(๐ดโŠ•(โˆ’๐ด๐‘‡)). Note also from Proposition 2.1 that ๐‘‹โˆˆ๐ถ๐ด if and only if ๐‘ฃ(๐‘‹)โˆˆIm(๐ดโŠ•(โˆ’๐ด๐‘‡)). Thus, ๐‘‹โˆˆ๐ถ๐ด if and only if ๐‘ฃ(๐‘‹) is a solution to the algebraic compatible linear system: ๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๐‘ฃ๎€ธ๎€ธ(๐‘‹)=๐‘ฃ(๐‘€)(A.9) for any ๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘› such that ๎€ท๎€ทrank๐ดโŠ•โˆ’๐ด๐‘‡๎€ท๎€ท๎€ธ๎€ธ=rank๐ดโŠ•โˆ’๐ด๐‘‡๎€ธ๎€ธ,๐‘ฃ(๐‘€)=๐‘›2โˆ’๐œˆ(0).(A.10) From Theorem 2.3, the nullity and the rank of ๐ดโŠ•(โˆ’๐ด๐‘‡) are, respectively, dimKer(๐ดโŠ•(โˆ’๐ด๐‘‡))=๐œˆ(0)rank(๐ดโŠ•(โˆ’๐ด๐‘‡))=๐‘›2โˆ’๐œˆ(0). Therefore, there exist permutation matrices ๐ธ,๐นโˆˆ๐‘๐‘›2ร—๐‘›2 such that there exists an equivalence transformation: ๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๎€ธโ‰ˆ๎€ท๎€ท๐ดโˆถ=๐ธ๐ดโŠ•โˆ’๐ด๐‘‡๎‚€๎€ธ๎€ธ๐น=Blockmatrix๐ด๐‘–๐‘—;๐‘–,๐‘—โˆˆ2๎‚(A.11) such that ๐ด11 is square nonsingular and of order ๐œˆ0. Define ๐‘€=๐ธ๐‘€๐นโ‰ˆ๐‘€(โ‰ 0)โˆˆ๐‘๐‘›ร—๐‘›. Then, the linear algebraic systems (๐ดโŠ•(โˆ’๐ด๐‘‡))๐‘ฃ(๐‘‹)=๐‘ฃ(๐‘€), and ๐ธ๎€ท๎€ท๐ดโŠ•โˆ’๐ด๐‘‡๎‚€๎€ธ๎€ธ๐น๐‘ฃ๐‘‹๎‚=๎ƒฌ๐ด11๐ด12๐ด21๐ด22๎ƒญโŽกโŽขโŽขโŽฃ๐‘ฃ๎‚€๐‘‹1๎‚๐‘ฃ๎‚€๐‘‹2๎‚โŽคโŽฅโŽฅโŽฆ=โŽกโŽขโŽขโŽฃ๐‘ฃ๎‚€๐‘€1๎‚๐‘ฃ๎‚€๐‘€2๎‚โŽคโŽฅโŽฅโŽฆ,๐‘‰๎‚€๐‘‹1๎‚=๐ดโˆ’111๎‚€๐‘ฃ๎‚€๐‘€1๎‚โˆ’๐ด12๐‘ฃ๎‚€๐‘‹2โŸบ๎‚€๎‚๎‚๐ด22โˆ’๐ด21๐ดโˆ’111๐ด12๎‚๐‘‰๎‚€๐‘‹2๎‚๎‚€=๐‘ฃ๐‘€2๎‚โˆ’๐ด21๐ดโˆ’111๐‘ฃ๎‚€๐‘€1๎‚(A.12) are identical if ๐‘‹ and ๐‘€ are defined according to ๐‘ฃ(๐‘‹)=๐น๐‘ฃ(๐‘‹) and ๐‘ฃ(๐‘€)=๐ธ๐‘ฃ(๐‘€). As a result, Properties (i) and (ii) follow directly from (A.12) for ๐‘€=๐‘€=0 and for any ๐‘€ satisfying rank(๐ดโŠ•(โˆ’๐ด๐‘‡))=rank(๐ดโŠ•(โˆ’๐ด๐‘‡),๐‘ฃ(๐‘€))=๐‘›2โˆ’๐œˆ(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 ๐€๐ถ pair wise commute and any arbitrary matrix commutes with itself (thus ๐‘—=๐‘– 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 [๐ด๐‘–,๐ด๐‘—]=[๐ด๐‘—,๐ด๐‘–]=0;โˆ€๐ด๐‘–,๐ด๐‘—โˆˆ๐€๐ถ what implies ๐ด๐‘–โˆˆ๐€๐ถ;โˆ€๐‘–โˆˆ๐‘โ‡”๐‘ฃ(๐ด๐‘–โ‹‚)โˆˆ๐‘–+1โ‰ค๐‘—โ‰ค๐‘Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—));โˆ€๐ด๐‘–,๐ด๐‘—โˆˆ๐€๐ถ.
(ii) It follows from its equivalence with Property (i) since Ker๐‘๐‘–(๐€๐ถโ‹‚)โ‰ก๐‘—(โ‰ ๐‘–)โˆˆ๐‘Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—)).
(iii) Property (iii) is similar to Property (i) for the whose set ๐‘€๐€๐ถ of matrices which commute with the set ๐€๐ถ so that it contains ๐€๐ถ and, furthermore, Ker๐‘(๐€๐ถโ‹‚)โ‰ก๐‘–โˆˆ๐‘Ker(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)).
(iv) It follows from โ‹ƒ๐‘—โˆˆ๐‘Im(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—))=โ‹‚๐‘—โˆˆ๐‘Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—));๐ด๐‘—โˆˆ๐€๐ถ and ๐‘๐‘›2โˆ‹0โˆˆKer(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—))โˆฉIm(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—)) but ๐‘๐‘›ร—๐‘›โˆ‹๐‘‹=0 commutes with any matrix in ๐‘๐‘›ร—๐‘› so that ๐‘๐‘›ร—๐‘›โˆ‹0โˆˆMC๐€๐ถโ‡”๐‘๐‘›ร—๐‘›โˆ‹0โˆ‰MC๐€๐ถ for any given ๐€๐ถ.
(v) and (vi) are similar to (ii)โ€“(iv) except that the members of ๐€ 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 ๐€๐ถ) implies that the above matrices ๐‘(๐€๐ถ),๐‘๐‘–(๐€๐ถ),๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—) are all rank defective and have at least identical number of rows than that of columns. Therefore, the square matrices ๐‘๐‘‡(๐€๐ถ)๐‘(๐€๐ถ),๐‘๐‘‡๐‘–(๐€๐ถ)๐‘๐‘–(๐€๐ถ), and ๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—) are all singular.

Proof of Theorem 3.3. (i) Any nonzero matrix ฮ›=diag(๐œ†๐œ†โ‹ฏ๐œ†), ๐œ†(โ‰ 0)โˆˆ๐‘ is such that ฮ›(โ‰ 0)โˆˆ๐ถ๐ด๐‘–(โˆ€๐‘–โˆˆ๐‘) so that ฮ›โˆˆ๐ถ๐€. Thus, 0โ‰ ๐‘ฃ(ฮ›)โˆˆKer๐‘(๐€)โ‡”๐‘›2>๐‘Ÿ๐‘Ž๐‘›๐‘˜๐‘(๐€)โ‰ฅrank๐‘๐‘–(๐€)โ‰ฅrank(๐ด๐‘–โŠ•(โˆ’๐ด๐‘–)); โˆ€๐‘–โˆˆ๐‘ and any given set ๐€. Property (i) has been proved.
(ii) The first part follows by contradiction. Assume โ‹‚๐‘–โˆˆ๐‘Ker(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–))={0} then 0โ‰ ๐‘ฃ(ฮ›)โˆ‰Ker๐‘(๐€) so that ฮ›=diag(๐œ†๐œ†โ‹ฏ๐œ†)โˆ‰๐ถ๐€, for any ๐œ†(โ‰ 0)โˆˆ๐‘ what contradicts (i). Also, ๐‘‹โˆˆ๐ถ๐ด๐‘–โ‡”๐‘ฃ(๐‘‹)โˆˆKer(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)); โˆ€๐‘–โˆˆ๐‘ so that ๐‘‹โˆˆ๐ถ๐€โ‹‚โ‡”๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘Ker(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)) what is equivalent to its contrapositive logic proposition ๐‘‹โˆˆ๐ถ๐€โ‹ƒโ‡”๐‘ฃ(๐‘‹)โˆˆ๐‘–โˆˆ๐‘Im(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)).
(iii) Let ๐€=๐€๐ถโ‡”๐ด๐‘–โˆˆ๐ถ๐ด๐‘—;โˆ€๐‘—(โ‰ ๐‘–)โˆˆ๐‘,โˆ€๐‘–โˆˆ๐‘โ‡”๐ด๐‘–โˆˆ๐ถ๐ด๐‘—;โˆ€๐‘—,๐‘–โˆˆ๐‘ since ๐ด๐‘–โˆˆ๐ถ๐ด๐‘–;โˆ€๐‘–โˆˆ๐‘๐‘ฃ๎€ท๐ด๐‘–๎€ธโˆˆ๎™๐‘–โˆˆ๐‘๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๎€ท๐ด๐‘โŸบ๐‘ฃ๐‘–๎€ธโˆˆ๎™๐‘–โˆˆ๐‘โงต{๐‘–}๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ธ๎€ธ;โˆ€๐‘–โˆˆ๐‘.(B.1) On the other hand, โŽ›โŽœโŽœโŽ๐‘ฃ๎€ท๐ด๐‘–๎€ธโˆˆ๎™๐‘—โˆˆ๐‘โงต๐‘–๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—๎€ท๐ด๎€ธ๎€ธโŸบ๐‘ฃ๐‘–๎€ธโˆˆ๐ถ๐ด๐‘—;โˆ€๐‘—โˆˆ๐‘โŽžโŽŸโŽŸโŽ forany๐‘–(<๐‘)โˆˆ๐‘.(B.2) This assumption implies directly that ๐‘ฃ๎€ท๐ด๐‘–๎€ธโˆˆ๐ถ๐ด๐‘—;โˆ€๐‘—โˆˆ๎€ท๐ด๐‘โˆง๐‘ฃ๐‘–+1๎€ธโˆˆ๎™๐‘—โˆˆ๐‘–+1๐ถ๐ด๐‘—forany๐‘–(<๐‘)โˆˆ๐‘(B.3) which together with ๐‘ฃ(๐ด๐‘–+1โ‹‚)โˆˆ๐‘—โˆˆ๐‘โงต๐‘–+1Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—)) implies that ๐‘ฃ๎€ท๐ด๐‘–+1๎€ธโˆˆ๐ถ๐ด๐‘—;โˆ€๐‘—โˆˆโŽ›โŽœโŽœโŽ๐‘ฃ๎€ท๐ด๐‘โŸน๐‘–+1๎€ธโˆˆ๎™๐‘—โˆˆ๐‘โงต๐‘–+1๎€ท๐ดKer๐‘—โŠ•๎€ทโˆ’๐ด๐‘‡๐‘—โŽžโŽŸโŽŸโŽ ๎€ธ๎€ธfor(๐‘–+1)โˆˆ๐‘.(B.4) Thus, it follows by complete induction that ๐€=๐€๐ถโ‡”๐‘ฃ(๐ด๐‘–โ‹‚)โˆˆ๐‘–โˆˆ๐‘โงต{๐‘–}Ker(๐ด๐‘—โŠ•(โˆ’๐ด๐‘‡๐‘—));โˆ€๐‘–โˆˆ๐‘ and Property (iii) has been proved.
(iv) The definition of M๐€๐ถ follows from Property (iii) in order to guarantee that [๐‘‹,๐ด๐‘–]=0; โˆ€๐ด๐‘–โˆˆ๐€. The fact that such a set contains properly ๐€๐ถโˆช{0} follows directly from ๐‘๐‘›ร—๐‘›โˆ‹ฮ›=diag(๐œ†๐œ†โ‹ฏ๐œ†)(โˆˆMC๐€๐ถ)โ‰ ๐€๐ถโˆช{0} for any ๐‘โˆ‹๐œ†โ‰ 0.

Proof of Theorem 3.4. If ๐ด๐‘–โˆผ๐ฝ๐ด๐‘–=๐‘ƒ๐‘–โˆ’1๐ด๐‘–๐‘ƒ๐‘–, with ๐ฝ๐ด๐‘– being the Jordan canonical form of ๐ด๐‘– then ๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–)โˆผ๐ฝ๐ด๐‘–โŠ•(โˆ’๐ฝ๐‘‡๐ด๐‘–)=๐‘‡๐‘–โˆ’1(๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–))๐‘‡๐‘– with ๐‘‡๐‘–=๐‘ƒ๐‘–โŠ—๐‘ƒ๐‘‡๐‘–โˆˆ๐‘๐‘›2ร—๐‘๐‘›2 (see proof of Theorem 2.3) being nonsingular; โˆ€๐‘–โˆˆ๐‘. Thus, (๐ด๐‘–โŠ•(โˆ’๐ด๐‘‡๐‘–))=๐‘‡๐‘–(๐ฝ๐ด๐‘–โŠ•(โˆ’๐ฝ๐‘‡๐ด๐‘–))๐‘‡๐‘–โˆ’1 so that: ๎€บ๐ด๐‘(๐€)=๐‘‡1โŠ•๎€ทโˆ’๐ด1๎€ธ๐ด๐‘‡2โŠ•๎€ทโˆ’๐ด2๎€ธโ‹ฏ๐ด๐‘‡๐‘โŠ•๎€ทโˆ’๐ด๐‘๎€ธ๎€ป๐‘‡=๎€บ๐ผ๐‘๐‘›2๐‘Šโˆ’๐‘ˆ๎€ป๎€บ๐‘‡1๐‘Š๐‘‡2๎€ป๐‘‡=๐“๐‰๐“๐š,(B.5) where ๎€บ๐‘‡๐“โˆถ=BlockDiag1๐‘‡2โ‹ฏ๐‘‡๐‘๎€ปโˆˆ๐‘๐‘๐‘›2ร—๐‘๐‘›2,๐“๐š๎€บ๐‘‡โˆถ=1โˆ’๐‘‡๐‘‡2โˆ’๐‘‡โ‹ฏ๐‘‡๐‘โˆ’๐‘‡๎€ป๐‘‡โˆˆ๐‘๐‘๐‘›2ร—๐‘›2,๎‚ƒ๐ฝ๐‰โˆถ=BlockDiag๐ด1โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด1๎‚๐ฝ๐ด2โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด2๎‚โ‹ฏ๐ฝ๐ด๐‘โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘๎‚๎‚„โˆˆ๐‘๐‘๐‘›2ร—๐‘๐‘›2.(B.6) Then, Ker๐‘(๐€)=๐‘๎™๐‘–=1๐ดKer๎€บ๎€ท๐‘–โŠ•๎€ทโˆ’๐ด๐‘‡๐‘–๎€ท๎€ธ๎€ธ๎€ป=Ker๐‰๐“๐š๎€ธโ‰ก๐‘๎™๐‘–=1๎‚€๐ฝKer๎‚ƒ๎‚€๐ด๐‘–โŠ•๎‚€โˆ’๐ฝ๐‘‡๐ด๐‘–๎€ท๐‘ƒ๎‚๎‚๐‘–โˆ’1โŠ—๐‘ƒ๐‘–โˆ’๐‘‡๎€ธ๎‚„๎‚(B.7) since ๐“ is nonsingular. Thus, โˆ€๐‘‹โˆˆDom(๐€)โŠ‚๐‘๐‘›2: ๐‘‹โˆˆ๐ถ๐€๎ƒฉโŸบ๐‘ฃ(๐‘‹)โˆˆ