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On the Block Structure and Frobenius Normal Form of Powers of Matrices
No full textThe Frobenius normal form of a matrix is an important tool in analyzing its properties. When a matrix is powered up, the Frobenius normal form of the original matrix and that of its powers need not be the same. In this article, conditions on a matrix and the power are provided so that for any invertible matrix , if is block upper triangular, then so is when partitioned conformably. The result is established for general matrices over any field. It is also observed that the contributions of the index of cyclicity to the spectral properties of a matrix hold over any field. The article concludes by applying the block upper triangular powers result to the cone Frobenius normal form of powers of a eventually cone nonnegative matrix
The cone of Z-transformations on Lorentz cone
No full textIn this paper, the structural properties of the cone of \calz-transformations on the Lorentz cone are described in terms of the semidefinite cone and copositive/completely positive cones induced by the Lorentz cone and its boundary. In particular, its dual is described as a slice of the semidefinite cone as well as a slice of the completely positive cone of the Lorentz cone. This provides an example of an instance where a conic linear program on a completely positive cone is reduced to a problem on the semidefinite cone
AE Regularity of Interval Matrices
No full textConsider a linear system of equations with interval coefficients, and each interval coefficient is associated with either a universal or an existential quantifier. The AE solution set and AE solvability of the system is defined by ∀∃- quantification. The paper deals with the problem of what properties must the coefficient matrix have in order that there is guaranteed an existence of an AE solution. Based on this motivation, a concept of AE regularity is introduced, which implies that the AE solution set is nonempty and the system is AE solvable for every right-hand side. A characterization of AE regularity is discussed, and also various classes of matrices that are implicitly AE regular are investigated. Some of these classes are polynomially decidable, and therefore give an efficient way for checking AE regularity. Eventually, there are also stated open problems related to computational complexity and characterization of AE regularity
Potentially Eventually Positive 2-generalized Star Sign Patterns
No full textA sign pattern is a matrix whose entries belong to the set . An -by- sign pattern is said to be potentially eventually positive if there exists at least one real matrix with the same sign pattern as and a positive integer such that for all . An -by- sign pattern is said to be potentially eventually exponentially positive if there exists at least one real matrix with the same sign pattern as and a nonnegative integer such that for all . Identifying necessary and sufficient conditions for an -by- sign pattern to be potentially eventually positive (respectively, potentially eventually exponentially positive), and classifying these sign patterns are open problems. In this article, the potential eventual positivity of the -generalized star sign patterns is investigated. All the minimal potentially eventually positive -generalized star sign patterns are identified. Consequently, all the potentially eventually positive -generalized star sign patterns are classified. As an application, all the minimal potentially eventually exponentially positive -generalized star sign patterns are identified. Consequently, all the potentially eventually exponentially positive -generalized star sign patterns are classified
Eigenvalue continuity and Gersgorin\u27s theorem
No full textTwo types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, the Ger\v{s}gorin disk theorem is revisited and the issue concerning the proofs of the theorem by continuity is clarified
Solution of symmetric positive semidefinite Procrustes problem
No full textIn this paper, the symmetric positive semidefinite Procrustes problem is considered. By using matrix inner product and matrix decomposition theory, an explicit expression of the solution is given. Based on the explicit expression given in this paper, it is easy to derive the explicit expression of the solution given by Woodgate [K.G. Woodgate. Least-squares solution of over positive semidefinite symmetric . {\em Linear Algebra Appl.}, 245:171--190, 1996.] and by Liao [A.P. Liao. On the least squares problem of a matrix equation. {\em J. Comput. Math.}, 17:589--594, 1999.] for the Procrustes problem in some special cases. The explicit expression given in this paper also shows that the solution of the special inverse eigenvalue problem considered by Zhang [L. Zhang. A class of inverse eigenvalue problem for symmetric nonnegative definite matrices. {\em J. Hunan Educational Inst.}, 2:11--17, 1995 (in Chinese).] can be computed exactly. Examples to illustrate the correctness of the theory results are given
Absolutely compatible pairs in a von Neumann algebra
No full textLet be elements in a unital C-algebra with . The element is absolutely compatible with if In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the closed unit ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form and with $|\alpha|^
m-nil-clean Companion Matrices
No full textCompanion matrices over fields of positive characteristic, , that are sums of idempotents, and a nilpotent are characterized in terms of dimension and trace of such a matrix and of $p.
Brauer\u27s theorem and nonnegative matrices with prescribed diagonal entries
No full textThe problem of the existence and construction of nonnegative matrices with prescribed eigenvalues and diagonal entries is an important inverse problem, interesting by itself, but also necessary to apply a perturbation result, which has played an important role in the study of certain nonnegative inverse spectral problems. A number of partial results about the problem have been published by several authors, mainly by H. \v{S}migoc. In this paper, the relevance of a Brauer\u27s result, and its implication for the nonnegative inverse eigenvalue problem with prescribed diagonal entries is emphasized. As a consequence, given a list of complex numbers of \v{S}migoc type, or a list with , and being realizable; and given a list of nonnegative real numbers , the remarkably simple condition is necessary and sufficient for the existence and construction of a realizing matrix with diagonal entries Conditions for more general lists of complex numbers are also given
