Linear Algebra and its Applications

Items from 1 to 20 out of 23 results

article

An asymptotic behavior of QR decomposition

Huajun Huang, Tin-Yau Tam

Linear Algebra and Its Applications > 2007 > 424 > 1 > 96-107

The m-th root of the diagonal of the upper triangular matrix R _m in the QR decomposition of AX ^m B=Q _m R _m converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A,B,X∈Cn×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of R _m is discussed. Some computational experiments are discussed.

article

Idempotency of linear combinations of three idempotent matrices, two of which are commuting

Oskar Maria Baksalary, Julio Benı´tez

Linear Algebra and Its Applications > 2007 > 424 > 1 > 320-337

The considerations of the present paper were inspired by Baksalary [O.M. Baksalary, Idempotency of linear combinations of three idempotent matrices, two of which are disjoint, Linear Algebra Appl. 388 (2004) 67–78] who characterized all situations in which a linear combination P=c1P1+c2P2+c3P3, with ci, i=1,2,3, being nonzero complex scalars and Pi, i=1,2,3, being nonzero complex idempotent matrices...

article

Classification of sesquilinear forms with the first argument on a subspace or a factor space

Vyacheslav Futorny, Vladimir V. Sergeichuk

Linear Algebra and Its Applications > 2007 > 424 > 1 > 282-303

Let V be a vector space over a field or skew field F, and let U be its subspace. We study the canonical form problem for bilinear or sesquilinear formsU×V→F,(V/U)×V→Fand linear mappings U→V, V→U, V/U→V, V→V/U. We solve it over F=C and reduce it over all F to the canonical form problem for ordinary linear mappings W→W and bilinear or sesquilinear forms W×W→F. Moreover, we give an algorithm that realizes...

article

Moving zeros among matrices

John Holbrook, Jean-Pierre Schoch

Linear Algebra and Its Applications > 2007 > 424 > 1 > 83-95

We investigate the zero-patterns that can be created by unitary similarity in a given matrix, and the zero-patterns that can be created by simultaneous unitary similarity in a given sequence of matrices. The latter framework allows a “simultaneous Hessenberg” formulation of Pati’s tridiagonal result for 4×4 matrices. This formulation appears to be a strengthening of Pati’s theorem. Our work depends...

article

Transition matrices for well-conditioned Markov chains

S.J. Kirkland, Michael Neumann, Jianhong Xu

Linear Algebra and Its Applications > 2007 > 424 > 1 > 118-131

Let T∈Rn×n be an irreducible stochastic matrix with stationary distribution vector π. Set A=I−T, and define the quantity κ3(T)≡12maxj=1,…,nπj‖Aj-1‖∞, where A _j , j=1,…,n, are the (n−1)×(n−1) principal submatrices of A obtained by deleting the jth row and column of A. Results of Cho and Meyer, and of Kirkland show that κ ₃ provides a sensitive measure of the conditioning of π under...

article

How real is your matrix?

Marko Huhtanen

Linear Algebra and Its Applications > 2007 > 424 > 1 > 304-319

For scalars there is essentially just one way to define reality, real part and to measure nonreality. In this paper various ways of defining respective concepts for complex-entried matrices are considered. In connection with this, products of circulant and diagonal matrices often appear and algorithms to approximate additively and multiplicatively with them are devised. Multiplicative structures have...

article

Extremal numbers of positive entries of imprimitive nonnegative matrices

Xingzhi Zhan

Linear Algebra and Its Applications > 2007 > 424 > 1 > 132-138

We determine the maximum and minimum numbers of positive entries of imprimitive nonnegative matrices with a given imprimitivity index. One application of the results is to estimate the imprimitivity index by the number of positive entries. The proofs involve the study of a cyclic quadratic form. This completes a research initiated by Lewin in 1990.

article

Photo of Roger Horn

Linear Algebra and Its Applications > 2007 > 424 > 1 > xii

article

Maps preserving the nilpotency of products of operators

Chi-Kwong Li, Peter Šemrl, Nung-Sing Sze

Linear Algebra and Its Applications > 2007 > 424 > 1 > 222-239

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ϕ:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ϕ(A)ϕ(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C⧹{0} such that one of the following holds:(a)There is a bijective bounded linear or conjugate-linear...

article

Mean matrices and infinite divisibility

Rajendra Bhatia, Hideki Kosaki

Linear Algebra and Its Applications > 2007 > 424 > 1 > 36-54

We consider matrices M with entries m _ij =m(λ _i ,λ _j ) where λ ₁ ,…,λ _n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w _ij =1/m (λ _i ,λ _j ) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied...

article

Schur product of matrices and numerical radius (range) preserving maps

Chi-Kwong Li, Edward Poon

Linear Algebra and Its Applications > 2007 > 424 > 1 > 8-24

Let F(A) be the numerical range or the numerical radius of a square matrix A. Denote by A∘B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F(A∘B)=F(ϕ(A)∘ϕ(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices.

article

Inequalities for sums and direct sums of Hilbert space operators

Omar Hirzallah, Fuad Kittaneh

Linear Algebra and Its Applications > 2007 > 424 > 1 > 71-82

We prove several singular value inequalities and norm inequalities involving sums and direct sums of Hilbert space operators. It is shown, among other inequalities, that if X and Y are compact operators, then the singular values of X+Y2 are dominated by those of X⊕Y. Applications of these inequalities are also given.

article

The spectral approximation of multiplication operators via asymptotic (structured) linear algebra

Stefano Serra-Capizzano

Linear Algebra and Its Applications > 2007 > 424 > 1 > 154-176

A multiplication operator on a Hilbert space may be approximated with finite sections by choosing an orthonormal basis of the Hilbert space. Multiplication operators with nonzero symbols, defined on L ² spaces of functions, are never compact and then such approximations cannot converge in the norm topology. Instead, we consider how well the spectra of the finite sections approximate the spectrum...

article

Eigenvalue inequalities for convex and log-convex functions

Jaspal Singh Aujla, Jean-Christophe Bourin

Linear Algebra and Its Applications > 2007 > 424 > 1 > 25-35

We give a matrix version of the scalar inequality f(a+b)⩽f(a)+f(b) for positive concave functions f on [0,∞). We show that Choi’s inequality for positive unital maps and operator convex functions remains valid for monotone convex functions at the cost of unitary congruences. Some inequalities for log-convex functions are presented and a new arithmetic–geometric mean inequality for positive matrices...

article

Geršgorin type theorems for quaternionic matrices

Fuzhen Zhang

Linear Algebra and Its Applications > 2007 > 424 > 1 > 139-153

This paper aims to set an account of the left eigenvalue problems for real quaternionic (finite) matrices. In particular, we will present the Geršgorin type theorems for the left (and right) eigenvalues of square quaternionic matrices. We shall conclude the paper with examples showing and summarizing some differences between complex matrices and quaternionic matrices and right and left eigenvalues...

article

Special issue dedicated to Roger Horn

Rajendra Bhatia, Fuad Kittaneh, Roy Mathias, Xingzhi Zhan

Linear Algebra and Its Applications > 2007 > 424 > 1 > 1-2

article

Sign patterns allowing nilpotence of index 3

Yubin Gao, Zhongshan Li, Yanling Shao

Linear Algebra and Its Applications > 2007 > 424 > 1 > 55-70

Suppose P is a property referring to a real matrix. We say that a sign pattern A allows P if there exists at least one matrix with the same sign pattern as A that has the property P. In this paper, we study sign patterns allowing nilpotence of index 3. Four methods for constructing sign patterns that allow nilpotence of index 3 are obtained. All tree sign patterns that allow nilpotence of index 3...

article

Lists of Editors

Linear Algebra and Its Applications > 2007 > 424 > 1 > ii-iii

article

Rank one operators and norm of elementary operators

Ameur Seddik

Linear Algebra and Its Applications > 2007 > 424 > 1 > 177-183

Let A be a standard operator algebra acting on a (real or complex) normed space E. For two n-tuples A=(A ₁ ,…,A _n ) and B=(B ₁ ,…,B _n ) of elements in A, we define the elementary operator R _A,B on A by the relation RA,B(X)=∑i=1nAiXBi for all X in A. For a single operator A∈A, we define the two particular elementary operators L _A and R _A on A...

article

On Euclidean distance matrices

R. Balaji, R.B. Bapat

Linear Algebra and Its Applications > 2007 > 424 > 1 > 108-117

If A is a real symmetric matrix and P is an orthogonal projection onto a hyperplane, then we derive a formula for the Moore–Penrose inverse of PAP. As an application, we obtain a formula for the Moore–Penrose inverse of an Euclidean distance matrix (EDM) which generalizes formulae for the inverse of a EDM in the literature. To an invertible spherical EDM, we associate a Laplacian matrix (which we...

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Volume:
424
Year:
2007

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Keywords

EIGENVALUE (3)
SPECTRAL RADIUS (2)
ALGEBRAICALLY CLOSED FIELDS (1)
BILINEAR AND SESQUILINEAR FORMS (1)
CANONICAL FORMS (1)
CANONICAL MATRICES (1)
CLASSIFICATION (1)
COMPACT OPERATOR (1)
CONDITION NUMBER (1)
CONJUGATION (1)
CONVEX FUNCTION (1)
CYCLE (1)
DCD-MATRIX (1)
DIFFERENTIABLE (1)
DIFFRACTIVE OPTICS (1)
DISTANCE MATRICES (1)
DOUBLY STOCHASTIC MATRIX (1)
EIGENVALUE OPTIMIZATION (1)
EIGENVALUES (1)
FOURIER BASIS (1)
FOURIER TRANSFORM (1)
GERŠGORIN THEOREM (1)
GROUP INVERSE (1)
HADAMARD PRODUCT (1)
HESSENBERG FORMS (1)
HIGHER-ORDER DERIVATIVE (1)
IMPRIMITIVE MATRIX (1)
INEQUALITY (1)
INFINITELY DIVISIBLE MATRICES (1)
INFINITELY DIVISIBLE MATRIX (1)
INVOLUTORY ANTIAUTOMORPHISMS (1)
LAPLACIAN (1)
LEFT EIGENVALUE (1)
LINEAR OPERATORS (1)
MAJORIZATION (1)
MARKOV CHAIN (1)
MATRIX OF QUATERNIONS (1)
MEAN (1)
MEASURE OF NONREALITY (1)
MULTIPLICATION OPERATOR (1)
NILPOTENCE (1)
NILPOTENT OPERATOR (1)
NONNEGATIVE MATRIX (1)
NUMERICAL RADIUS (1)
NUMERICAL RANGE (1)
OBLIQUE PROJECTOR (1)
OPERATOR MONOTONE FUNCTION (1)
ORTHOGONAL POLYNOMIALS (1)
ORTHOGONAL PROJECTOR (1)
PARTITIONED MATRIX (1)
PERTURBATION THEORY (1)
POSITIVE DEFINITE MATRIX (1)
PRESERVER (1)
PRODUCTS OF CIRCULANT AND DIAGONAL MATRICES (1)
QR DECOMPOSITION (1)
QUADRATIC FORM (1)
QUATERNION (1)
QUATERNIONIC MATRICES (1)
QUATERNIONIC MATRIX (1)
RIGHT EIGENVALUE (1)
SCHUR PRODUCT (1)
SIGN PATTERN (1)
SINGULAR EIGENVALUE (1)
SINGULAR VALUE (1)
SKEWSYMMETRIC MATRICES (1)
SPECTRAL FUNCTION (1)
SPECTRUM (1)
STATIONARY DISTRIBUTION (1)
STOCHASTIC MATRIX (1)
SYMBOL (1)
SYMMETRIC FUNCTION (1)
SYMMETRIC MATRICES (1)
TENSOR ANALYSIS (1)
TOEPLITZ (AND GENERALIZED LOCALLY TOEPLITZ) SEQUENCES (1)
TREE SIGN PATTERN (1)
TREES (1)
TRIDIAGONAL MATRICES (1)
TWICE DIFFERENTIABLE (1)
UNITAL POSITIVE LINEAR MAP (1)
UNITARILY INVARIANT NORM (1)
UNITARY SIMILARITY (1)
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Linear Algebra and its Applications

An asymptotic behavior of QR decomposition

Idempotency of linear combinations of three idempotent matrices, two of which are commuting

Classification of sesquilinear forms with the first argument on a subspace or a factor space

Moving zeros among matrices

Transition matrices for well-conditioned Markov chains

How real is your matrix?

Extremal numbers of positive entries of imprimitive nonnegative matrices

Photo of Roger Horn

Maps preserving the nilpotency of products of operators

Mean matrices and infinite divisibility

Schur product of matrices and numerical radius (range) preserving maps

Inequalities for sums and direct sums of Hilbert space operators

The spectral approximation of multiplication operators via asymptotic (structured) linear algebra

Eigenvalue inequalities for convex and log-convex functions

Geršgorin type theorems for quaternionic matrices

Special issue dedicated to Roger Horn

Sign patterns allowing nilpotence of index 3

Lists of Editors

Rank one operators and norm of elementary operators

On Euclidean distance matrices

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Linear Algebra and its Applications

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