on left eigenvalues of a quaternionic matrix abstract in this paper we introduce the concept of left and right eigenvalues for a quaternionic matrix and investigate their properties quantities and relationship

determinantal representation of the generalized inverses over the quaternion skew field with applications we first present some determinantal representations of one 1 5 inverse of a quaternion matrix within the framework of a theory of the row and column determinants as applications we show some new explicit expressions of generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field finally we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations the findings of this paper extend some known results in the literature

determinantal representation of weighted generalized inverses abstract in this article we introduce new determinantal representations of the weighted generalized inverse by using a lemma of robinson d w robinson the classical adjoint linear algebra appl 411 2005 254 276 and using generalized cramer s rule we get the method for computing minors of the weighted moore penrose inverse and the w weighted drazin inverse

non markovian quantum trajectories instruments and time continuous measurements the linear and nonlinear non markovian quantum state diffusion nmqsd equations are well known tools for the description of certain non markovian open quantum systems in this work we systematically investigate whether the normalized linear nmqsd or the nonlinear nmqsd solutions can be generated by means of a time continuous measurement by considering any conceivable measurement scheme we derive a necessary criterion for the measurement interpretation of both equations concrete examples show that the normalized linear nmqsd solutions are realizable only in the markovian limit in general the application of the presented criterion to the nonlinear nmqsd remains an open issue furthermore this criterion is a necessary one not only in the context of the nmqsds but also for a much broader class of stochastic schrodinger equations

a characterization for the w weighted drazin inverse and a cramer rule for the w weighted drazin inverse solution we establish characterization for the w weighted drazin inverse of an arbitrary rectangular matrix which reduces to the well known result if the matrix is nonsingular also a cramer rule for finding the unique w weighted drazin inverse solution x r aw k 1 of special restricted linear equationswawx b b r wa k 2 is presented and reduces to the classical cramer rule if a is invertible

right eigenvalues for quaternionic matrices a topological approach abstract we apply the lefschetz fixed point theorem to show that every square matrix over the quaternions has right eigenvalues we classify them and discuss some of their properties such as an analogue of jordan canonical form and diagonalization of elements of the compact symplectic group sp n

integral representation of the w weighted drazin inverse two integral representations are presented for the w weighted drazin inverse a d w of rectangular matrices a w the first one has some restriction on the eigenvalues of aw or wa the second one holds without any restriction on the eigenvalues

determinantal representations of the w weighted drazin inverse over the quaternion skew field within the framework of the theory of the column and row determinants we obtain new determinantal representations of the w weighted drazin inverse over the quaternion skew field we give determinantal representations of the w weighted drazin inverse by using previously introduced determinantal representations of the drazin inverse the moore penrose inverse and the limit representations of the w weighted drazin inverse in some special case

a note on the perturbation of the w weighted drazin inverse the perturbation bound is presented for the w weighted drazin inverse for a rectangular matrix under the weaker condition the application to the singular linear equations is also given in this note

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

a bound of lengths of chains of minimal rational curves on fano manifolds of picard number 1 abstract in this paper we investigate the minimal length of chains of minimal rational curves needed to join two general points on a fano manifold of picard number 1 under mild assumptions in particular we give a sharp bound of the length by a fundamental argument as an application we compute the length for fano manifolds of dimension 7

explicit formulas for determinantal representations of the drazin inverse solutions of some matrix and differential matrix equations the drazin inverse solutions of the matrix equations ax b xa b and axb d are considered in this paper we use both the determinantal representations of the drazin inverse obtained earlier by the author and in the paper we get analogs of the cramer rule for the drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations x ax b and x xa b where the matrix a is singular

the boundedness of degree of fano varieties with picard number one contents 0 introduction 1 rational curves 1 1 deformation theory of morphisms 1 2 free rational curves 1 3 the invariant d m 2 special subvarieties in fano varieties of large degree 2 1 the statement 2 2 a siegel type lemma 2 3 multiplicity subspaces 2 4 the product theorem part i 2 5 the product theorem part ii 2 6 the proof 3 the covering lemma 4 collections of subsets 4 1 base change and push forward 4 2 an example 4 3 universally open morphisms 4 4 flat families 5 proof of theorem a

lengths of chains of minimal rational curves on fano manifolds abstract in this paper we consider a natural question how many minimal rational curves are needed to join two general points on a fano manifold x of picard number 1 in particular we study the minimal length of such chains in the cases where the dimension of x is at most 5 the coindex of x is at most 3 and x equips with a structure of a double cover as an application we give a better bound on the degree of fano 5 folds of picard number 1

rational curves of minimal degree and characterizations of projective spaces in this paper we investigate complex uniruled varieties x whose rational curves of minimal degree satisfy a special property namely we assume that the tangent directions to such curves at a general point x x form a linear subspace of txx as a first application of our main result we give a unified geometric proof of mori s wahl s campana peternell s and andreatta wi niewski s characterizations of open image in new window we also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree