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Let A and B be two generalized quadratic matrices with respect to idempotent matrices P and Q , respectively, such that ( A − α P ) ( A − β P ) = 0 , A P = P A = A , ( B − γ Q ) ( B − δ Q ) = 0 , B Q = Q B = B , P Q = Q P , AB ≠ BA , and ( A + B ) ( α β P − γ δ Q ) = ( α β P − γ δ Q ) ( A + B ) with α , β , γ , δ ∈ C . Let A + B be diagonalizable. The relations between the spectrum of the matrix A + B and the spectra of some matrices produced from A and B are considered. Moreover, some results on the spectrum of the matrix A + B are obtained when A + B is not diagonalizable. Finally, some results and examples illustrating the applications of the results in the work are given. [ABSTRACT FROM AUTHOR]

Consider the space M n nor of square normal matrices X = ( x i j ) over R ∪ { − ∞ } , i.e., − ∞ ≤ x i j ≤ 0 and x i i = 0 . Endow M n nor with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A ∈ M n nor and consider the set Ω ( A ) of matrices in M n nor which commute with A . We prove that Ω ( A ) is a finite union of alcoved polytopes; in particular, Ω ( A ) is a finite union of convex sets. The set Ω A ( A ) of X such that A ⊙ X = X ⊙ A = A is also a finite union of alcoved polytopes. The same is true for the set Ω ′ ( A ) of X such that A ⊙ X = X ⊙ A = X . A topology is given to M n nor . Then, the set Ω A ( A ) is a neighborhood of the identity matrix I . If A is strictly normal, then Ω ′ ( A ) is a neighborhood of the zero matrix. In one case, Ω ( A ) is a neighborhood of A . We give an upper bound for the dimension of Ω ′ ( A ) . We explore the relationship between the polyhedral complexes span A , span X and span ( A X ) , when A and X commute. Two matrices, denoted A ̲ and A ¯ , arise from A , in connection with Ω ( A ) . The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension. [ABSTRACT FROM AUTHOR]

This paper deals with the idempotency, involution and nilpotency of two matrices. In this paper, by introducing a new idea into the classical block technique, we characterize all situations in which a linear combination of the formis(Q1) square-zero when P and Q are both idempotent;(Q2) square-zero when P is tripotent and Q is idempotent, respectively;(Q3) idempotent when P is tripotent and Q is involutory, respectively;(Q4) involutory when P is tripotent and Q is involutory, respectively.In addition, we point out several other related problems that the reformed method in this paper applies. [ABSTRACT FROM AUTHOR]

Letandbenonzero quadratic matrices and letandbe nonzero complex numbers. Necessary and sufficient conditions for the quadraticity of the linear combinations of the formare obtained. Our main results contain many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices. Finally, some numerical examples are given to exemplify the main results. [ABSTRACT FROM AUTHOR]

Let R denote the Galois ring of characteristic p s and cardinality p s h . In this paper, we determine the Smith normal forms of matrices over R , compute the number of the orbits of m × n matrices under the group GL m ( R ) × GL n ( R ) and the length of each orbit. Moreover, we discuss their applications to association schemes, systems of linear equations, generalized invertible matrices, idempotent matrices and involutory matrices. [ABSTRACT FROM AUTHOR]

In this paper, we study the Drazin inverse of P+Q, where P is idempotent, and derive additive formulas under condition PQ² = 0 or Q²P = 0. As its applications we establish some representations for the Drazin inverse of a class of block matrices with an idempotent subblock and under some conditions expressed in terms of the individual blocks. The results extend earlier work obtained by several authors. [ABSTRACT FROM AUTHOR]

A new representation of AT,S(2,3) of A having prescribed range space T and null space S is derived. Using this representation, the well known generalized inverses such as the Moore–Penrose inverse, the group inverse, the Drazin inverse, the Bott–Duffin inverse and the generalized Bott–Duffin inverse are computed. Three numerical examples are worked out to demonstrate the computation of the Moore–Penrose inverse, the Drazin inverse and the {2,3}-inverse to show the efficacy of our approach. [ABSTRACT FROM AUTHOR]

Abstract: A ring is called nil-clean if each of its elements is a sum of an idempotent and a nilpotent. In response to a question of S. Breaz et al. in [1], we prove that the matrix ring over a division ring D is a nil-clean ring if and only if . As consequences, it is shown that the matrix ring over a strongly regular ring R is a nil-clean ring if and only if R is a Boolean ring, and that a semilocal ring R is nil-clean if and only if its Jacobson radical is nil and is a direct product of matrix rings over . [Copyright &y& Elsevier]

Abstract: We consider the problem of characterizing situations where a linear combination of commuting involutive matrices is a k-potent matrix, where k is a positive integer such that . Also, we consider the problem of characterizing situations where a linear combination of the form is a nonsingular or an involutive matrix, when idempotent matrices satisfy different conditions and . [Copyright &y& Elsevier]

Abstract: For given matrices F and G over an arbitrary field , necessary and sufficient conditions (in terms of rank, amongst others) are presented for F to divide G, where either F itself is idempotent or one of its quotients is idempotent, or both. Formulae are also given by which these quotients can be constructed. Finally, new proofs of some known results on idempotent factorization are presented in the present context to provide another perspective on the behaviour of such factorizations. [Copyright &y& Elsevier]

After a survey of some known lattice results, we determine the greatest idempotent (resp. compact) solution, when it exists, of a finite square rational equation assigned over a linear lattice. Similar considerations are presented for composite relational equations.

Abstract: This paper examines matrices such that where , the identity matrix, and where s and k are nonnegative integers with . Spectral theory is used to characterize these matrices. The cases s =0 and are considered separately since they are analyzed by different techniques. [Copyright &y& Elsevier]

A number of characterizations involving column and null spaces of various functions of a pair of oblique projectors are established. Particular attention is paid to properties of a product of the two projectors, but other functions of the pair (such as sum, difference or sum and difference of products) are considered as well. In many cases, the results obtained generalize those already known to be valid for orthogonal projectors. [ABSTRACT FROM AUTHOR]

The relations between the spectrum of the matrix Q+ R and the spectra of matrices (γ+δ)Q+(α+β)R-QR-RQ, QR-RQ, αβR - QRQ, αRQR - (QR)², and βR - QR have been given assuming that the matrix Q + R is diagonalizable, where Q, R are {α,β}-quadratic matrix and {γδ}-quadratic matrix, respectively, of order n. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, -potent matrices are considered. A matrix is called -potent when where K is an involutory matrix and . Specifically, -potent matrices are analyzed considering their relations to different classes of complex matrices. These classes of matrices are: -generalized projectors, -Hermitian matrices, normal matrices, and matrices (anti-)commuting with K or such that KB is involutory, Hermitian or normal. In addition, some new relations for K-generalized centrosymmetric matrices have been derived. [Copyright &y& Elsevier]

Abstract: This paper concerns with the tripotency of a linear combination of three matrices, which has a background in statistical theory. We demonstrate all the possible cases that lead to the tripotency of a linear combination of two involutory matrices and a tripotent matrix that mutually commute. By utilizing block technique and returning the partitioned matrices to their original forms, we derive the sufficient and necessary conditions such that a linear combination of three mutually commuting involutory matrices is tripotent. [Copyright &y& Elsevier]

Abstract: It has been established a -term disjoint idempotent decomposition (DID) for the linear combinations produced from n commutative tripotent matrices, their products and their products of power 2 at most. The results obtained in this way generalize those in [Y. Tian, A disjoint idempotent decomposition for linear combinations produced from two commutative tripotent matrices and its applications, Linear Multilinear Algebra 59 (2011) 1237–1246]. Moreover, an algorithm to get a DID has been provided. Finally, a numerical example has been given to exemplify the results. [Copyright &y& Elsevier]

Abstract: Let be an arbitrary field, and be elements of such that the polynomials and are split in . Given a square matrix , we give necessary and sufficient conditions for the existence of two matrices A and B such that , and . Prior to this paper, such conditions were known in the case , and and in the case . Here, we complete the study, which essentially amounts to determining when a matrix is the sum of an idempotent and a square-zero matrix. This generalizes results of Wang to an arbitrary field, possibly of characteristic 2. [Copyright &y& Elsevier]

Abstract: In this paper, the sufficient and necessary conditions for the existence of the group inverse for a class of 2×2 anti-triangular block matrices M with some certain conditions satisfied, and representations of the group inverse of M are obtained, which can be regarded as the extension of Bu et al. [C. Bu, J. Zhao, J. Zheng, Group inverse for a class 2×2 block matrices over skew fields, Appl. Math. Comput. 204 (2008) 45–49] and [C. Bu, J. Zhao, K. Zhang, Some results on group inverses of block matrices over skew fields, Electron. J. Linear Algebra, 18 (2009) 117–125]. Further, we use these results to determine the expressions of the Drazin inverse of M. Finally, two numerical examples are given to illustrate our results. [Copyright &y& Elsevier]

As is well known, every positive idempotent matrix is of rank 1. It is proved that idempotent matrices without zeros have this property over many skew semifields, and all these skew semifields are described. [ABSTRACT FROM AUTHOR]

Abstract: Recently, situations where a matrix coincides with some of its powers have been studied. This kind of matrices is related to the generalized inverse matrices. On the other hand, it is possible to introduce another class of matrices that involve an involutory matrix, generalizing the well-known idempotent matrix, widely useful in many applications. In this paper, we introduce a new kind of matrices called -potent, as an extension of the aforementioned ones. First, different properties of -potent matrices have been developed. Later, the main result developed in this paper is the characterization of this kind of matrices from a spectral point of view, in terms of powers of the matrix, by means of the group inverse and, via a block representation of a matrix of index 1. Finally, an application of the above results to study linear combinations of -potent matrices is derived. [Copyright &y& Elsevier]

It is well known after R. Swan that K͊0(R[X,Y, Z]/(X² + Y² + Z² - 1)) is isomorphic to the integers &#2124, whenever R is a field of characteristic not two which contains the squared root of -1, see [6, Corollary 10.8] and [7, §7]. First, we give explicit idempotent matrices γp of order two, corresponding to the integer p, in the isomorphism above, if R is a field of characteristic zero. Then, we use the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of R[X,Y, Z]/(X⊃2 +Y⊃2 + Z⊃2 - 1) to itself, and relate the problem of finding hermitian representatives for R = K(i), K a field not containing i, to some unsolved problems of representing Brouwer degrees by polynomial maps [8]. [ABSTRACT FROM AUTHOR]

A square matrix A of order n is said to be tripotent if A 3 = A. In this note, we give a nine-term disjoint idempotent decomposition for the linear combination of two commutative tripotent matrices and their products. Using the decomposition, we derive some closed-form formulae for the eigenvalues, determinant, rank, trace, power, inverse and group inverse of the linear combinations. In particular, we show that the linear combinations of two commutative tripotent elements and their products can produce 39 = 19,683 tripotent elements. [ABSTRACT FROM PUBLISHER]

Abstract: We investigate simultaneous solutions of the matrix Sylvester equations , where and are k-tuples of commuting matrices of order and , respectively. We show that the matrix Sylvester equations have a unique solution X for every compatible k-tuple of matrices if and only if the joint spectra and are disjoint. We discuss the connection between the simultaneous solutions of Sylvester equations and related questions about idempotent matrices separating disjoint subsets of the joint spectrum, spectral mapping for the differences of commuting k-tuples, and a characterization of the joint spectrum via simultaneous solutions of systems of linear equations. [Copyright &y& Elsevier]

Abstract: Let and be two complex matrices satisfying and . For nonzero complex numbers such that is diagonalizable, we relate the spectrum of to the spectra of , , and . [Copyright &y& Elsevier]

A square matrix A of order n is said to be involutory if A2 = In, and to be idempotent if A2 = A. In this article, we give two universal similarity factorization equalities for linear combinations of two commutative involutory and two idempotent matrices and their products. As applications, we derive some disjoint decompositions for these linear combinations, and use the disjoint decompositions to derive a variety of results on the determinants, ranks, traces, inverses, generalized inverses and similarity decompositions of these linear combinations. In particular, we present some collections of involutory, idempotent and tripotent matrices generated from these linear combinations. [ABSTRACT FROM AUTHOR]

Abstract: For , the matrix is said to be the Drazin inverse of A, if it holds that , where r is the smallest nonnegative integer such that . In this paper, we give the representation for the Drazin inverse of the block matrix . [ABSTRACT FROM AUTHOR]

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈  and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ. [ABSTRACT FROM AUTHOR]

complex square matrix A is called an orthogonal projector if A = A = A*, where A* denotes the conjugate transpose of A. In this paper, we give a comprehensive investigation to matrix expressions consisting of orthogonal projectors and their properties through ranks of matrices. We first collect some well-known rank formulas for orthogonal projectors and their operations, and then establish various new rank formulas for matrix expressions composed by orthogonal projectors. As applications, we derive necessary and sufficient conditions for various equalities for orthogonal projectors and their operations to hold. [ABSTRACT FROM AUTHOR]

The paper researches the rank of combinations a PA + b AQ- c PAQ of two idempotent matrices P and Q. Using the properties of the idempotent matrix and elementary block matrix operation, we get some rank equalities for combinations a PA + b AQ- c PAQ of two idempotent matrices P and Q. These rank equalities generalize the results of Koliha J J, Rakočević V and Tian Y, and give some applications of the rank equalities. [ABSTRACT FROM AUTHOR]

Abstract: and Trenkler pointed out that if a difference of idempotent matrices P and Q is nonsingular, then so is their sum, and Koliha et al expressed explicitly a condition, which combined with the nonsingularity of ensures the nonsingularity of . In the present paper, these results are strengthened by showing that the nonsingularity of is in fact equivalent to the nonsingularity of any combination (where ), and the nonsingularity of is equivalent to the nonsingularity of any combination (where ). [Copyright &y& Elsevier]

Abstract: We study the idempotent matrices over a commutative antiring. We give a characterization of idempotent matrices by digraphs. We study the orbits of conjugate action and find the cardinality of orbits of basic idempotents. Finally, we prove that invertible, linear and idempotent preserving operators on matrices over entire antirings are exactly conjugate actions for . We also give a complete characterization of the case. [Copyright &y& Elsevier]

Abstract: In their paper [Y. Tian, G.P.H. Styan, Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001) 101–117], Tian and Styan established several rank equalities involving a pair of idempotent matrices P and Q. Subsequently, these results are reinvestigated from the point of view of the following question: provided that idempotent P, Q are Hermitian, which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces? Simultaneously, some related results are established, which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors. [Copyright &y& Elsevier]

Abstract: The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which and commute, Linear and Multilinear Algebra 14 (1984) 241–256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings. [Copyright &y& Elsevier]

Abstract: Let and be two nonzero commuting n × n tripotent matrices and two nonzero complex numbers. Necessary and sufficient conditions for the tripotency and the idempotency of are obtained. The problems considered here have also statistical importance when are real scalars and are real symmetric matrices. [Copyright &y& Elsevier]

The problem of characterizing all situations in which aA + bB is an idempotent matrix when A2 = A, Bk + 1 = B, AB ≠ BA, and a, b are nonzero complex numbers is studied. [ABSTRACT FROM AUTHOR]