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Rank and dimension functions
No full textIn this paper, we invoke theory of generalized inverses and minus partial order on regular matrices over a commutative ring to define rankâfunction for regular matrices and dimensionâfunction for finitely generated projective modules which are direct summands of a free module. Some properties held by the rank of a matrix and the dimension of a vector space over a field are generalized. Also, a generalization of rank-nullity theorem has been established when the matrix given is regular
Real nullstellensatz and *-ideals in *-algebras
No full textFor a fixed tuple of square matrices X ={X_1,...,X_g} the set I(X) of all noncommutative polynomials p in X and Xâ such that p(X) = 0 is an ideal in the â-algebra of all polynomials. This article concerns such zeroes and their corresponding ideals. An algebraic characterization of ideals of the form I(X) is a real nullstellensatz. A main result of this article is a strong nullstellensatz for a â-ideal of finite codimension in a â-algebra. Without the finite codimension assumption, there are examples of such ideals which do not satisfy, very liberally interpreted, any Nullstellensatz. A polynomial p in noncommuting variables (x_1,...,x_g,xâ_1,...,x_âg) is called analytic if it is a polynomial in the variables x_j only. As shown in this article, â-ideals generated by analytic polyno-mials do satisfy a natural Nullstellensatz and those generated by homogeneous analytic polynomials have a particularly simple description. Another natural notion of zero of a noncommutative polynomial p is a pair (X, v) such that p(X)v = 0; here X is an n by n matrix tuple and v â R^n. For fixed (X,v), the set of all such polynomials is a left ideal. The relationship between such zeroes and their left ideals is considerably more developed than is our beginning effort here. This article provides a guide to that literature
Pseudo Schur complements, pseudo principal pivot transforms and their inheritance properties
No full textExtensions of the Schur complement and the principal pivot transform, where the usual inverses are replaced by the Moore-Penrose inverse, are revisited. These are called the pseudo Schur complement and the pseudo principal pivot transform, respectively. First, a generalization of the characterization of a block matrix to be an M-matrix is extended to the nonnegativity of the Moore-Penrose inverse. A comprehensive treatment of the fundamental properties of the extended notion of the principal pivot transform is presented. Inheritance properties with respect to certain matrix classes are derived, thereby generalizing some of the existing results. Finally, a thorough discussion on the preservation of left eigenspaces by the pseudo principal pivot transformation is presented
On (T,f)-connections of matrices and generalized inverses of linear operators
No full textIn this note, generalized connections Ï_{T ,f} are investigated, where AÏ_{T ,f} B = T_Af(T_A)^â(B) for positive semidefinite matrix A and hermitian matrix B, and operator monotone function f : J â R on an interval J â R. Here the symbol (T_A)^â denotes a reflexive generalized inverse of a positive bounded linear operator T_A. The problem of estimating a given generalized connection by other ones is studied. The obtained results are specified for special cases of α-arithmetic, α-geometric and α-harmonic operator means
Spectral Bounds for Matrix Polynomials with Unitary Coefficients
No full textIt is well known that the eigenvalues of any unitary matrix lie on the unit circle. The purpose of this paper is to prove that the eigenvalues of any matrix polynomial, with unitary coefficients, lie inside the annulus A_{1/2,2) := {z â C | 1/2 < |z| < 2}. The foundations of this result rely on an operator version of Roucheâs theorem and the intermediate value theorem
Maxima of the signless Laplacian spectral radius for planar graphs
No full textThe signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian. In this paper, we prove that the graph has the maximal signless Laplacian spectral radius among all planar graphs of order
Minimization problems for certain structured matrices
No full textFor given Z, B â C^{n\times k}, the problem of finding A â C^{n\times n}, in some prescribed class W, that minimizes ||AZ â B|| (Frobenius norm) has been considered by different authors for distinct classes W. Here, this minimization problem is studied for two other classes, which include the symmetric Hamiltonian, symmetric skew-Hamiltonian, real orthogonal symplectic and unitary conjugate symplectic matrices. The problem of minimizing ||A â AË||, where AË is given and A is a solution of the previous problem, is also considered (as has been done by others, for different classes W). The key idea of this contribution is the reduction of each one of the above minimization problems to two independent subproblems in orthogonal subspaces of C^{n\times n}. This is possible due to the special structures under consideration. Matlab codes are developed, and numerical results of some tests are presented
On Wilkinson's problem for matrix pencils
No full textSuppose that an n-by-n regular matrix pencil A -\lambda B has n distinct eigenvalues. Then determining a defective pencil Eâ\lambda F which is nearest to Aâ\lambda B is widely known as Wilkinsonâs problem. It is shown that the pencil E â\lambda F can be constructed from eigenvalues and eigenvectors of A â\lambda B when A â \lambda B is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A â\lambda B to E â \lambdaF is the minimum âgapâ between the eigenvalues of A â \lambdaB. As a consequence, lower and upper bounds for the âWilkinson distanceâ d(L) from a regular pencil L(\lambda) with distinct eigenvalues to the nearest non-diagonalizable pencil are derived.Furthermore, it is shown that d(L) is almost inversely proportional to the condition number of the most ill-conditioned eigenvalue of L(\lambda)
A strange phenomenon for the singular values of commutators with rank one matrices
No full textThe singular values of XY ô Y X are the objects under investigation. Here, X andY are square matrices with complex entries, and one of them has rank one. Hence, there are at most two non-trivial numbers among the commutator's singular values, and the pairs of interest can be depicted in the plane. The emphasis will lie on the unexpectedly intriguing case in which both matrices are of rank one { because the result then is astonishingly complex, and the problem gives rise to interpretations unveiling geometry acting in the background
Solving the real eigenvalues of hermitian quadratic eigenvalue problems via bisection
No full textThis paper considers solving the real eigenvalues of the Quadratic Eigenvalue Problem (QEP) Q(\lambda)x =(\lambda^2M+\lambdaC+K)x = 0 in a given interval (a, b), where the coefficient matrices M, C, K are Hermitian and M is nonsingular. First, an inertia theorem for the QEP is proven, which characterizes the difference of inertia index between Hermitian matrices Q(a) and Q(b). Several useful corollaries are then obtained, where it is shown that the number of real eigenvalues of QEP Q(\lambda)x = 0 in the interval (a, b) is no less than the absolute value of the difference of the negative inertia index between Q(a) and Q(b); furthermore, when all real eigenvalues in (a, b) are semi-simple with the same sign characteristic, the inequality becomes an equality. Based on the established theory, the bisection method (with preprocessing) can be used to compute the real eigenvalues of the QEP by computing the inertia indices. Applications to the calculation of the equienergy lines with k.p model, and also a non-overdamped mass-spring system are presented in the numerical tests