on angles and distances between subspaces abstract the angles and distances between two given subspaces of c n 1 are investigated on the basis of a joint decomposition of the corresponding orthogonal projectors several new results are established with the particular attention paid to the notions of inclinedness and minimal angle to demonstrate the usefulness of the approach utilized some results known to be valid in hilbert space are reestablished in c n 1 either in generalized form or with considerably shorter proofs than in the original sources

on disjoint range matrices abstract for a square complex matrix f and for f being its conjugate transpose the class of matrices satisfying r f r f 0 where r denotes range column space of a matrix argument is investigated besides identifying a number of its properties several functions of f such as f f f f ff f f and f f are considered particular attention is paid to the moore penrose inverses of those functions and projectors attributed to them it is shown that some results scattered in the literature whose complexity practically prevents them from being used to deal with real problems can be replaced with much simpler expressions when the ranges of f and f are disjoint furthermore as a by product of the derived formulae one obtains a variety of relevant facts concerning for instance rank and range

constructing projections on sums and intersections abstract in this note we list several formulas for computing orthogonal projections onto the linear sum and intersection of two subspaces of c n in terms of the projections on the individual subspaces

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

a positivstellensatz for non commutative polynomials a non commutative polynomial which is positive on a bounded semi algebraic set of operators has a weighted sum of squares representation this positivstellensatz parallels similar results in the commutative case a broader issue is to what extent does real semi algebraic geometry extend to non commutative polynomials our strict positivstellensatz is positive news on the opposite extreme from strict positivity would be a real nullstellensatz we give an example which shows that there is no non commutative real nullstellensatz along certain lines however we include a successful type of non commutative nullstellensatz proved by george bergman

inheritance principles for chordal graphs abstract we prove two results concerning positive completions of partial positive matrices first we give a parametrization of all positive invertible completions of a partial positive matrix having as support a chordal graph along a fixed increasing chordal sequence then a formula for the determinant of each completion is obtained in terms of the associated parameters as an application we obtained a proof of a conjecture of w w barrett c r johnson and l rodman regarding an inheritance principle for chordal graphs

a generalized companion matrix of a polynomial and some applications abstract a generalized companion matrix of a polynomial is introduced the concept is applied to estimates of polynomial roots and to the explicit determination of the characteristic equation of matrix

positive semidefinite quadratic forms on unitary matrices abstract let f x 1 x n i j 1 n ij x i x j a ij a ji r be a real quadratic form such that the trace of the hermitian matrix f v 1 v n i j 1 n ij v i v j is nonnegative for all unitary 2 n 2 n matrices v 1 v n we prove that f u 1 u n is positive semidefinite for all unitary matrices u 1 u n of arbitrary size m m

matrix methods for polynomials abstract for every complex polynomial a set of special matrices with the characteristic polynomial p will be given this enables us to apply the gerschgorin theorem

linear construction of companion matrices this note is concerned with the following problem for a given matrix a e c ix and a vector a e c does there exist a mapping 2 assigning to each manic polynomial f of degree n a vector x f e c such that the matrix b a a x f is a companion matrix off i e the characteristic polynomial of b is t l ftf the classes of suitable matrices a and vectors a are characterized and some properties of b are described the corresponding unique mapping is determined by a system of linear equations the cases of a triangular bidiagonal or diagonal matrix a are discussed explicitly and many known companion matrices are obtained as particular cases then gershgorin s theorem is applied yielding error estimates for polynomial roots finally the extension to block companion matrices and an example of nonlinear construction are discussed

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

on a subspace metric based on matrix rank abstract the metric between subspaces m n c n 1 defined by m n rk p m p n where rk denotes rank of a matrix argument and p m and p n are the orthogonal projectors onto the subspaces m and n respectively is investigated such a metric takes integer values only and is not induced by any vector norm by exploiting partitioned representations of the projectors several features of the metric m n are identified it turns out that the metric enjoys several properties possessed also by other measures used to characterize subspaces such as distance also called gap frobenius distance direct distance angle or minimal angle

column space equalities for orthogonal projectors 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

factorization of operator valued polynomials in several non commuting variables abstract a version of fejer riesz factorization and factorization of positive operator valued polynomials in several non commuting variables holds the proofs use arveson s extension theorem and matrix completions

structure and entropy for positive definite toeplitz kernels on free semigroups abstract in this article we study the structure of positive definite toeplitz kernels on free semigroups called also multi toeplitz and its implications in noncommutative dilation theory harmonic analysis on fock spaces prediction and interpolation theory for stationary stochastic processes a parametrization of positive definite multi toeplitz kernels in terms of generalized schur sequences of contractions is obtained this leads to explicit minimal naimark dilations cholesky factorizations szego type limit theorems and entropy for positive definite multi toeplitz kernels maximal outer factors are associated with positive definite multi toeplitz kernels and are used to obtain inner outer factorization for operators on fock spaces with coefficients in hilbert spaces the kolmogorov wiener prediction problem for stochastic processes having as covariance kernels positive definite multi toeplitz kernels is considered the predication error operator is calculated in terms of schur parameters resp maximal outer factor associated with the covariance kernel of the process and a connection with a szego type infimum problem is established we solve the caratheodory interpolation problem for positive definite multi toeplitz kernels we obtain a parametrization of all solutions in terms of schur sequences and we find the maximal entropy solution the results of this article can be used to develop a theory of stochastic n linear systems

extending positive definiteness the main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of semigroups to positive definite ones by specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems like complex two sided complex and multidimensional trigonometric moment problems in addition unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme

eigenvalues of functions of orthogonal projectors by representing two orthogonal projectors in a finite dimensional vector space as partitioned matrices several characterizations concerning eigenvalues of various functions of the pair are obtained these results substantially extend the ones already available in the literature additionally some related results dealing with the functions of a pair of orthogonal projectors are provided with the emphasis laying on the problem of invertibility

on a pair of vector spaces the paper provides an extensive investigation of a pair of vectors spaces with a particular attention paid to perpendicular subspaces several new properties of the pair are identified some of which enabled to derive known facts about subspaces by means of simpler proofs the approach utilized exploits partitioned representations of orthogonal projectors and is based on the fact that there is one to one correspondence between an orthogonal projector and the subspace onto which it projects

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

positive noncommutative polynomials are sums of squares hilbert s 17th problem concerns expression of polynomials on r n as a sum of squares it is well known that many positive polynomials are not sums of squares see re d a for excellent surveys in this paper we consider symmetric noncommutative polynomials and call one matrix positive if whenever matrices of any size are substituted for the variables in the polynomial the matrix value which the polynomial takes is positive semidefinite the result in this paper is a polynomial is matrix positive if and only if it is a sum of squares

on the projectors ff and f f abstract a particular version of the singular value decomposition is exploited for an extensive analysis of two orthogonal projectors namely ff and f f determined by a complex square matrix f and its moore penrose inverse f various functions of the projectors are considered from the point of view of their nonsingularity idempotency nilpotency or their relation to the known classes of matrices such as ep bi ep gp dr or sr this part of the paper was inspired by benitez and rakocevic j benitez v rakocevic matrices a such that aa a a are nonsingular appl math comput 217 2010 3493 3503 further characteristics of ff and f f with a particular attention paid on the results dealing with column and null spaces of the functions and their eigenvalues are derived as well besides establishing selected exemplary results dealing with ff and f f the paper develops a general approach whose applicability extends far beyond the characteristics provided therein

completions of matrices and the commutant lifting theorem abstract completion results are obtained for partially defined matrices with an infinite number of entries as a corollary the commutant lifting theorem of sz nagy and foia is proved by solving a completion problem

extensions of positive definite functions on free groups abstract an analogue of krein s extension theorem is proved for operator valued positive definite functions on free groups the proof gives also the parametrization of all extensions by means of a generalized type of szego parameters one singles out a distinguished completion called central which is related to quasi multiplicative positive definite functions an application is given to factorization of noncommutative polynomials