Halim Özdemir

On linear combinations of two tripotent, idempotent, and involutive matrices

Abstract Let A = c 1 A 1 + c 2 A 2 , where c 1 , c 2 are nonzero complex numbers and ( A 1 , A 2 ) is a pair of two n × n nonzero matrices. We consider the problem of characterizing all situations where a linear combination of the form A = c 1 A 1 + c 2 A 2 is (i) a tripotent or an involutive matrix when A 1 and A 2 are commuting involutive or commuting tripotent matrices, respectively, (ii) an idempotent matrix when A 1 and A 2 are involutive matrices, and (iii) an involutive matrix when A 1 and A 2 are involutive or idempotent matrices.

Generalized quadraticity of linear combination of two generalized quadratic matrices

It is investigated the necessary and sufficient conditions for the generalized quadraticity of a linear combination of any two generalized quadratic matrices. The main result obtained is, in a sense, a generalization of the main results given in [Uç M, Özdemir H, Özban AY. On the quadraticity of linear combinations of quadratic matrices. Linear Multilinear Algebra. 2015;63:1125–1137.] which contains many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices, to the generalized quadratic matrices.

On the quadraticity of linear combinations of quadratic matrices

Let and be nonzero quadratic matrices and let and be nonzero complex numbers. Necessary and sufficient conditions for the quadraticity of the linear combinations of the form are 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.

On the spectra of some combinations of two generalized quadratic matrices

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 - ? Q ) = 0 , B Q = Q B = B , P Q = Q P , AB ? BA, and ( A + B ) ( α β P - γ ? Q ) = ( α β P - γ ? Q ) ( A + B ) with α , β , γ , ? ? 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.

On the best approximate (P,Q)-orthogonal symmetric and skew-symmetric solution of the matrix equation AXB=C

Abstract - Suppose that the matrix equation AXB = C with unknown matrix X is given, where A, B, and C are known matrices of suitable size. The matrix nearness problem is considered over the (P,Q)-orthogonal symmetric and (P,Q)-orthogonal skew-symmetric solutions of the matrix equation AXB=C. The explicit forms of the best approximate solutions of the problems considered are established. Moreover, two numerical examples and a comparative table, depending on the examples chosen from literature, are given.

The generalized quadraticity of linear combinations of two commuting quadratic matrices

Let and be two nonzero commuting and -quadratic matrices, respectively, where with and . The aim of this work is mainly to characterize all situations, where the linear combination is a generalized quadratic matrix. The results established here cover many of the results in the literature related to idempotency, involutivity and tripotency of the linear combinations of idempotent and/or involutive matrices.

On the BLUEs in Two Linear Models via C. R. Rao's Pandora's Box

We consider the estimation of the parameters in two partitioned linear models, denoted by 𝒜 = {y, X 1 β 1 + X 2 β 2, V 𝒜} and ℬ = {y, X 1 β 1 + X 2 β 2, V ℬ}, which we call full models. Correspondingly, we define submodels 𝒜1 = {y, X 1 β 1, V 𝒜} and ℬ1 = {y, X 1 β 1, V ℬ}. Using the so-called Pandoras Box approach introduced by Rao (1971, we give new necessary and sufficient conditions for the equality between the best linear unbiased estimators (BLUEs) of X 1 β 1 under 𝒜1 and ℬ1 as well as under 𝒜 and ℬ. In our considerations we will utilise the Frisch–Waugh–Lovell theorem which provides a connection between the full model 𝒜 and the reduced model 𝒜 r = {M 2 y, M 2 X 1 β 1, M 2 V 𝒜 M 2} with M 2 being an appropriate orthogonal projector. Moreover, we consider the equality of the BLUEs under the full models assuming that they are equal under the submodels.

On a disjoint idempotent decomposition for linear combinations produced from n commutative tripotent matrices

Abstract It has been established a 3 n -term disjoint idempotent decomposition (DID) for the linear combinations produced from n ( ⩾ 2 ) 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.

BİR GENEL PARÇALANMIŞ LiNEER MODEL ve İLİŞKİLİ İNDİRGENMİŞ LİNEER MODELLER ALTINDA TAHMİNLERİN KARŞILAŞTIRILMASI

Bir genel parcalanmis lineer modele kars ilik gelen indirgenmis lineer model altinda gozlenebilir rasgele vektorun beklenen degerinin BLUE sunun, genel parcalaninis lineer model altinda da BLUE kalmasi icin gerek ve yeter bir kosul detayli olarak incelenmektedir.