on the continuity of the map ϑ ϑ from the predual of a w algebra abstract let be elements in the predual of a w algebra for their absolute value parts the estimate 2 1 2 is obtained

inequality between powers of positive semidefinite matrices abstract for p 1 we present reasonable nonnegative functions f t on 0 such that in the space of matrices 0 a b and b f a imply b p a p

on the extension of linear contractions some results of the hahn banach type resume let be a norm on rn and s a compact convex semigroup of linear contractions given two k tuples of n vectors x 1 x k and y 1 y k we seek conditions for the existence of a contraction s s that simultaneously takes x i to y i that is sx i y i for all i 1 k straightforward application of the separation theorem for convex sets provides a general but abstract result in the form of a system of inequalities specializing to the 1 norm and the norm respectively and s to comprise either all contractions or those contractions that preserve a particular linear form it is possible to evaluate the characteristic functionals arising from the separation theoem therby the abstract results can be rduced to a tractable form which turns out to be of the hahn banach type

numerical integration of related hankel transforms by quadrature and continued fraction expansion an algorithm is presented for the accurate evaluation of hankel or bessel transforms of algebraically related kernel functions defined here as the non bessel function portion of the integrand that is more widely applicable than the standard digital filter methods without enormous increases in computational burden the algorithm performs the automatic integration of the product of the kernel and bessel functions between the asymptotic zero crossings of the latter and sums the series of partial integrations using a continued fraction expansion equivalent to an analytic continuation of the series the integrands may be saved to allow the rapid computation of related transforms without recalculating the kernel or bessel functions and the integration steps use interlacing quadrature formulas so that no function evaluations are wasted when it is necessary to increase the order of the quadrature rule the continued fraction algorithm allows very slowly divergent or even formally divergent integrals to be computed quite easily the local error is controlled at each step in the algorithm and accuracy is limited largely by machine resolution the algorithm is written in fortran and is listed in an appendix along with a driver program that illustrates its features the driver program and subroutine are available from the seg business offtce

isospectral matrices and classical polynomials abstract the matrices of order n defined in terms of the n arbitrary numbers x j by the formulae x diag x j and z jk jk l 1 n x j x l 1 1 jk x j x k 1 are representations of the multiplicative operator and of the differential operator d d in a space spanned by the polynomials in of degree less than n this elementary fact implies a number of remarkable formulae involving these matrices including novel representations of the classical polynomials

the schur complements of generalized doubly diagonally dominant matrices as is known the schur complements of diagonally dominant matrices are diagonally dominant the same is true of doubly diagonally dominant matrices the purpose of this paper is to extend the results to the generalized doubly diagonally dominant matrices a proper subset of h matrices that is we show that the schur complement of a generalized doubly diagonally dominant matrix is a generalized doubly diagonally dominant matrix

spectral inequalities for matrix exponentials abstract this note generalizes an inequality of bernstein as follows if c is an n n complex matrix and c k is the k th compound of c 1 k n n n k and if the eigenvalues of c k are labeled in order of decreasing magnitude 1 c k 2 c k n c k define the partial trace tr k i c by tr k i c h 1 i h c k i 1 n then for any complex n n matrix a tr k i e a e a tr k i e a a i 1 n with equality if a is normal or k n a spectral inequality of k fan is also generalized through the use of compound matrices

imaging low frequency and dc electromagnetic fields using a simple linear approximation we consider that all types of electromagnetic measurements represent weighted averages of the subsurface electrical conductivity distribution and that to each type of measurement there corresponds a different weighting function we use this concept for the quantitative interpretation of dc resistivity magnetometric resistivity and low frequency electric and magnetic measurements at low induction numbers in all three cases the corresponding inverse problems are nonlinear because the weighting functions depend on the unknown conductivity distribution we use linear approximations that adapt to the data and do not require reference resistivity values the problem is formulated numerically as a solution of a system of linear equations the unknown conductivity values are obtained by minimizing an objective function that includes the quadratic norm of the residuals as well as the spatial derivatives of the unknowns we also apply constraints through the use of quadratic programming the final product is the flattest model that is compatible with the data under the assumption of the given weighting functions this approximate inversion or imaging technique produces reasonably good results for low and moderate conductivity contrasts we present the results of inverting jointly and individually different data sets using synthetic and field data

triangular truncation and finding the norm of a hadamard multiplier for b a fixed matrix we study the problem of using a criterion of haagerup to find the norm of the map a a b where is the hadamard or entrywise product of matrices the techniques developed are applied to triangular truncation and it is proved that if kn is the norm of triangular truncation of n n matrices then knlogn 1

2 5d controlled source em modeling with general 3d source geometries most 2 5d controlled source electromagnetic csem modeling algorithms presented to date explicitly consider only sources that are point dipoles oriented parallel or perpendicular to the direction of constant conductivity this makes simulations of complex source geometries expensive requiring separate evaluations of many point dipole fields and thus limits the practical applicability of such schemes for simulating and interpreting field data we present a novel 2 5d csem modeling scheme that overcomes this limitation and permits efficient simulations of sources with general shape and orientation by evaluating fields for the entire source at once we accommodate general sources by using a secondary field approach in which primary fields are computed for the general source and a 1d background conductivity model to carry out the required fourier transforms between space and wavenumber domain using the same fast cosine and sine transform filters as in conventional algorithms we split the primary and secondary fields into their symmetric and antisymmetric parts for complex 3d source geometries this approach is significantly more efficient than previous 2 5d algorithms our finite difference algorithm also includes novel approaches for divergence correction at low frequencies and em field interpolation across conductivity discontinuities we describe the modeling scheme and demonstrate its accuracy and efficiency by comparisons of 2 5d simulated data with 1d and 3d results

an extremal problem in fourier analysis with applications to operator theory a certain minimal extrapolation problem for fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators in this paper we estimate the value of the constant in the fourier transform problem by an analytic reformulation

a simple sensitivity analysis of time domain and frequency domain electromagnetic measurements electromagnetic responses in both time and frequency domains can be looked upon as weighted spatial averages of the nearby conductivity distribution using this concept i analyze the simplest possible situation a uniform half space in the presence of uniform current sources both transient and harmonic measurements are equally effective in providing conductivity values for the half space while the conductivity values are the same the averaging processes are not to different types of measurements even within the same domain there correspond different types of sensitivity or weighting functions some of these functions emphasize the near surface conductivity while others emphasize the conductivity at depth these and other features of the weighting functions explain the behavior of magnetotelluric sounding curves

transient electromagnetic inversion a remedy for magnetotelluric static shifts surficial bodies can severely distort magnetotelluric mt apparent resistivity data to arbitrarily low frequencies this distortion known as the mt static shift is due to an electric field generated from boundary charges on surficial inhomogeneities and persists throughout the entire mt recording range static shifts are manifested in the data as vertical parallel shifts of log log apparent resistivity sounding curves the impedance phase being unaffected using a three dimensional 3 d numerical modeling algorithm simulated mt data with finite length electrode arrays are generated significant static shifts are produced in this simulation however for some geometries they are impossible to identify techniques such as spatial averaging and electromagnetic array profiling emap are effective in removing static shifts but they are expensive especially for correcting a previously collected mt data set parametric representation and use of a single invariant quantity such as the impedance tensor

a subclass of symmetric loewner matrices abstract the subclass of real symmetric loewner matrices with fixed interpolation nodes and row sums zero is studied it is shown that it forms a commutative semigroup with respect to multiplication and contains with each element its moore penrose inverse as well

electromagnetic modeling of 3 d sources over 2 d inhomogeneities in the time domain a numerical method is presented by which the transient electromagnetic response of a two dimensional 2 d conductor embedded in a conductive host rock and excited by a rectangular current loop can be modeled this 2 5 d modeling problem has been formulated in the time domain in terms of a vector diffusion equation for the scattered magnetic induction which is fourier transformed into the spatial wavenumber domain in the strike direction of the conductor to confine the region of solution of the diffusion equation to the conductive earth boundary values for the components of the magnetic induction on the ground surface have been calculated by means of an integral transform of the vertical component of the magnetic induction at the air earth interface the system of parabolic differential equations for the three magnetic components has been integrated for 9 to 15 discrete spatial wavenumbers ranging from 10 7 to 4 10 2m 1 using an implicit homogeneous finite difference scheme the discretization of the

holder type norm inequalities for schur products of matrices abstract for a unitarily invariant norm on m n and p 1 we define a p by a p 1 p then p is again a unitarily invariant norm we give holder type inequalities for schur products of the form a b 0 p0 a 1 p1 b 2 p2 as a corollary we settle in a stronger form a conjecture of marcus et al on submultiplicativity of a unitarily invariant norm with respect to schur multiplication we prove also holder type inequalities for operator radii w of the form w a b w a w b

a generalization of a a 1 i abstract a matrix inequality is obtained in an elementary way for the schur product of two positive definite matrices the inequality generalizes a well known result due to fiedler which asserts that if a is a positive definite matrix then a a 1 i is a positive semidefinite matrix where denotes the schur multiplication

on the norm of a schur product abstract if a x is the schur product of n n matrices a and x then we study estimates on the norm of the map x a x where x has the norm it acquires as a linear operator on a complex n dimensional hilbert space

majorization between the diagonal elements and the eigenvalues of an oscillating matrix abstract in this note we give an answer to an open question posed by marshall and olkin on the majorization between the diagonal elements and the eigenvalues of an oscillating matrix

convergence analysis of high order algebraic fractional step schemes for time dependent stokes equations in this paper we analyze the family of yosida algebraic fractional step schemes proposed in a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 f saleri and a veneziani siam j numer anal 43 2005 pp 174 194 and p gervasio f saleri and a veneziani j comput phys 214 2006 pp 347 365 when applied to time dependent stokes equations under suitable regularity assumptions on the data splitting error estimates both for velocity and pressure are established in particular we analyze the first three methods of this family providing respectively convergence of the fractional step solution towards the numerical solution achieved without any operator splitting of orders 3 2 5 2 7 2 for the velocity and 1 2 3 for the pressure moreover a general way to set up higher order schemes is proposed the present analysis is carried out when spectral element methods are employed for space discretization

induced norms of the schur multiplier operator abstract fix an n by n complex matrix a and consider the operator x sa x a x on n by n complex matrices x where a x denotes the schur product of a and x we show that the induced norm of sa with respect to the numerical radius norm is at most one if and only if the matrix a admits a factorization a b wb where w is a contractive matrix and the euclidean norms of the columns of b are at most one we give other equivalent characterizations and derive as a consequence a formally similar result about the induced norm of sa with respect to the spectral norm

an operator inequality abstract for hilbert space operators with s invertible hermitian it is proved that sts 1 s 1 ts 2 t

furuta s inequality and its application to ando s theorem abstract as a continuation of preceding notes we discuss furuta s inequality under the chaotic order defined by log a log b for positive invertible operators a and b which is applied to a generalization of ando s theorem consequently we obtain furuta s inequality under the chaotic order

schur products and matrix completions we prove that a necessary and sufficient condition for a given partially positive matrix to have a positive completion is that a certain schur product map defined on a certain subspace of matrices is a positive map by analyzing the positive elements of this subspace we obtain new proofs of the results of h dym and i gohberg and grone johnson sa and wolkowitz linear algebra appl 58 1984 109 124 linear algebra appl 36 1981 1 24 we also obtain a new proof of a result of u haagerup decomposition of completely bounded maps on operation algebras preprint characterizing the norm of schur product maps and a new hahn banach type extension theorem for these maps finally we obtain generalizations of many of these results to matrices of operators which we apply to the study of representations of certain subalgebras of the n n matrices

schur complements and statistics abstract in this paper we discuss various properties of matrices of the type s h ge 1 f which we call the schur complement of e in a e f g h the matrix e is assumed to be nonsingular when e is singular or rectangular we consider the generalized schur complements s h ge f where e is a generalized inverse of e a comprehensive account of results pertaining to the determinant the rank the inverse and generalized inverses of partitioned matrices and the inertia of a matrix is given both for schur complements and for generalized schur complements we survey the known results in a historical perspective and obtain several extensions numerous applications in numerical analysis and statistics are included the paper ends with an exhaustive bibliography of books and articles related to schur complements

direct inversion of time domain electromagnetic data this note describes a simple method for converting transient electromagnetic em sounding data into profiles of conductivity versus depth based on an approximate image representation for the decaying induced ground currents the method can provide one dimensional 1 d inversion for any kind of time domain em sounding data the discussion here is limited to the case of central loop sounding in particular i apply the inversion to some time derivative central loop sounding data and demonstrate that essentially the same interpretation of the geoelectric section is obtained with the simple image method as is obtained using traditional iterative least squares fitting to layered models this simple inverse can be computed much more quickly than an iterative least squares inverse making it possible to estimate the geoelectric section concomitant with data acquisition

matrices differential operators and polynomials it is noted that the matrix z of order n defined in terms of the n arbitrary numbers xj by the formula zjk jk j l 1 xj xl 1 1 jk xj xk 1 may be considered in an appropriate framework to correspond to the differential operator d dx there follow prescriptions to construct explicit matrices of arbitrary order n in terms of n or more arbitrary numbers or of the n roots of given polynomials matrices whose eigenvalues and eigenvectors are given fully or in part by very simple formulas novel representations of the classical polynomials hermite laguerre lagrange gegenbauer jacobi are also obtained such as the formula for hermite polynomials hn x 2n det xi h where i is the unit matrix of order n and the matrix h of order n is defined by hjk 2n 1 2 jk n 1 exp 2i j 1 2 exp 2i j 1 jk exp 2i j 2i sin j k 1 exp i j k with j j n arbitrary

a quantitative version of the observation that the hadamard product is a principal submatrix of the kronecker product abstract the hadamard and kronecker products of two n m matrices a b are related by a b p t 1 a b p 2 where p 1 p 2 are partial permutation matrices after establishing several properties of the p matrices this relationship is employed to demonstrate how a simplified theory of the hadamard product can be developed during this process the well known result a b a b aa bb is extended to a b a b 1 2 aa bb ab ba aa bb showing an inherent link between the hadamard product and conventional product of two matrices this leads to a sharper bound on the spectral norm of a b a b 1 2 a 2 b 2 ab 2 1 2 a b and an improvement on the weak majorization of a b 2 a b w 1 2 2 a 2 b 2 ab w 2 a 2 b for a real non singular matrix x and invertible diagonal matrices d e the spectral condition number is shown to be if scaled bounded below as follows dxe 2 x x t 2 x x t 2 1 2 x x t 1 for a 0 we have i a 2 1 2 i a 2 a a i a 2 and a 1 2 a 1 2 2 1 2 i a a 1 a a 1 when a 0 the latter inequality is compared to styan s inequality a a 1 1 2 i a a 1 when a is a correlation matrix and is shown to possess stronger properties of ordering finally the relationship a b p t 1 a b p 2 is applied to determine conditions of singularity of certain orderings of the hadamard products of matrices

the use and misuse of apparent resistivity in electromagnetic methods problems and misunderstandings arise with the concept of apparent resistivity when the analogy between an apparent resistivity computed from geophysical observations and the true resistivity structure of the subsurface is drawn too tightly several definitions of apparent resistivity are available for use in electromagnetic methods however those most commonly used do not always exhibit the best behavior many of the features of the apparent resistivity curve which have been interpreted as physically significant with one definition disappear when alternative definitions are used it is misleading to compare the detection or resolution capabilities of different field systems or configurations solely on the basis of the apparent resistivity curve for the in loop transient electromagnetic tem method apparent resistivity computed from the magnetic field response displays much better behavior than that computed from the induced voltage response a comparison of exact and asymptotic formulas for the te

on a matrix inequality abstract let s n be the set of all n n real symmetric matrices we give a complete characterization of the set of matrix convex functions defined on s n we also consider connections between matrix monotonicity in the lowner sense and matrix convexity

analysis of spectral variation and some inequalities a geometric method based on a decomposition of the space of complex matrices is employed to study the variation of the spectrum of a matrix when adapted to special cases this leads to some classical inequalities as well as some new ones as an example of the latter we show that if u v are unitary matrices and k is a skew hermitian matrix such that uv exp k then for every unitary invariant norm the distance between the eigenvalues of v and those of vis bounded by k this generalises two earlier results which used particular unitary invariant norms

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

the golden thompson trace inequality is complemented we prove a class of trace inequalities which complements the golden thompson inequality for example tr epa epb 2 p tr ea b holds for all p 0 when a and b are hermitian matrices and denotes the geometric mean we also prove related trace inequalities involving the logarithmic function namely p 1tr x log yp 2xpyp 2 tr x log x log y p 1tr x log xp 2ypxp 2 for all p 0 when x and y are nonnegative matrices these inequalities supply lower and upper bounds on the relative entropy

spectral variation of normal matrices abstract it has been a durable conjecture that the distance appropriately defined between the spectra of two normal matrices is bounded by the operator norm of their difference we report on some numerical studies that show that this conjecture is in general false in fact even in the 3 x 3 case the spectral distance may exceed the norm distance by between 1 and 2 percent we also describe the background of the conjecture and offer an analysis of extremal counterexamples

a generalization of the hoffman wielandt theorem abstract the well known inequality of a j hoffman and h w wielandt is extended from single normal operators to commuting tuples of such operators

an inequality for hilbert schmidt norm for the absolute value c c c 1 2 and the hilbert schmidt norm c hs trc c 1 2 of an operatorc the following inequality is proved for any bounded linear operatorsa andb on a hilbert space a b hs leqq 2 1 2 a b hs the corresponding inequality for two normal state and of a von neumann algebram is also proved in the following form d varphi psi leqq xi varphi xi psi leqq 2 1 2 d varphi psi here denotes the unique vector representative of a state in a natural positive conep form andd denotes the bures distance defined as the infimum which is also the minimum of the distance of vector representatives of and in particular xi varphi 1 xi varphi 2 leqq 2 1 2 xi 1 xi 2 for any vector representatives j of j j 1 2

an automatic technique for presentation of coincident loop impulse response transient electromagnetic data coincident loop tem sounding data are often presented by plotting the half space apparent conductivity as a function of delay time a new algorithm generates an improved presentation that plots the apparent conductivity as a function of depth the resulting data may be further processed to sharpen or spike the smoothly varying apparent conductivity depth curves in an attempt to better represent the rapid changes in conductivity that often exist in the earth the algorithm described involves an approximation but is simple easy to use and computationally efficient a layered conductivity structure is assumed so the algorithm is best for areas where the geology is approximately horizontal however the algorithm can also be used to identify anomalous features that are not infinite horizontal layers the spiked conductivity models derived from synthetic data are consistent with the original layered earth models and show a greater resolution than the apparent conductivity depth curves and sometimes ampl

on some operator inequalities a real valued continuous function f 2 defined on a finite or infinite interval of the real line is said to be operator monotone if for any pair of bounded selfadjoint operators a b on an infinite dimensional hilbert space a b implies f a f b here f a and f b are defined according to the usual functional calculus and refers to the order relation induced by the cone of positive semi definite operators operator monotony is quite a restrictive condition in fact for p 0 the function f 2 2 p is operator monotone on the half line 0 c if and only if 0 1 the function f 2 2 p possesses some property akin to operator monotony in this direction furuta i 3 recently observed that if r 0 p 0 q 1 satisfies 1 2r q p 2r then a b 0 implies a tp 2 q arbpa r l if p 2r and q 2 the requirement is automatically satisfied hence a p ap 2bpap 2 i 2 whenever a b 0 when a is invertible this means that

block matrix generalizations of schur s basic theorems on hadamard products abstract in a classic 1911 paper i schur gave several useful bounds for the spectral norm and eigenvalues of the hadamard entrywise product of two matrices motivated by applications to the theory of monotone and convex matrix functions we are led to consider hadamard products in which both factors are conformally partitioned block matrices and the entries of one factor are constant within each block such products are special cases of a block kronecker tensor product and it is in this context that we present generalizations of schur s results

modeling of borehole resistivity measurements using infinite electrical grids an infinite electrical resistive grid generated by a finite difference approximation to poisson s equation in cylindrical coordinates with no azimuthal variation has been used to model borehole resistivity measurements the medium surrounding the borehole is infinite in extent and its resistivity varies only in the radial direction as a result the branch resistance values of the infinite grid also vary only in the radial direction a continued fraction of laurent operators determines the node voltages along the borehole s surface use of an operational calculus based upon the finite fourier transformation converts the continued fraction into a readily computed form this yields new formulas for apparent resistivity given various resistivity measuring arrays and a resistivity profile in the radial direction in contrast to classical methods the need to match boundary conditions at the interface of a cylindrically layered earth is eliminated so that the procedure is just as easily applied to a continuo

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

inverses of 2 2 block matrices abstract in this paper the authors give explicit inverse formulae for 2 2 block matrices with three different partitions then these results are applied to obtain inverses of block triangular matrices and various structured matrices such as hamiltonian per hermitian and centro hermitian matrices

majorization and singular values iii abstract we obtain a majorization inequality which relates the singular values of a complex square matrix a and those of t a where t is a linear map the inequality takes a particularly simple form if t is positive the results generalize several known majorization inequalities for eigenvalues and singular values

coincident loop transient electromagnetic master curves for interpretation of two layer earths a set of apparent conductivity master curves has been calculated for the coincident loop transient electromagnetic tem method used over a two layer earth conductivity contrasts range from 0 001 to 1000 loop radius layer depth ratios range from 0 01 to 100 the time range is sufficient to see the entire shape of the curves from the early to the late time asymptotes these curves allow the determination of the parameters of a two layer earth for accurate data over a sufficient time range examples using the curves to interpret multilayered earths are given the curves are also used to show the limitations placed on interpretation by existing tem equipment

conductivity depth imaging of airborne electromagnetic step response data an adaptation of the macnae lamontagne method allows transform of airborne step response electromagnetic em data to a conductivity depth image the algorithm is based on a nonlinear transformation of the amplitude of the measured response at each delay time to an apparent mirror image depth using matrix algebra the set of mirror image depth delay time data pairs can then be used to derive a conductivity section data can be efficiently processed on a personal computer at rates faster or comparable to the rate required for collection stable conductivity fitting as a function of depth is obtained by damping the matrix inversion by specification of the first and second derivative smoothness weights of the fitted conductivity depth sounding damping parameters may be either fixed or varied along the profile their choice can be constrained by geologic control stability of the process is enhanced by accounting for the transmitter and receiver tilts the mirror image depth delay time data can also be use

symmetric function means of positive operators abstract many authors have studied means involving ratios of the elementary symmetric functions this work involves generalizing these means to an operator theoretic setting and extending many of the classical scalar inequalities to operators it is shown that these operator means are related to electrical networks

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

a note on the hadamard product of matrices it is shown that the smallest eigenvalue of the hadamard product a b of two positive definite hermitian matrices is bounded from below by the smallest eigenvalue of abt

a generalization of spectral radius numerical radius and spectral norm abstract let c n be the linear space of complex column vectors with n coordinates associated with the usual inner product denote by k the set of all k tuples of orthonormal vectors in c n given a real number p 1and an n n complex matrix a with eigenvalues 1 n we define p k max j 1 k i j p 1 p 1 i 1 k n r p k max i 1 k x i ax i p 1 p x 1 x k k and a p k max i 1 k x i ay i p 1 p x 1 x k y 1 y k k as the p k spectral radius p k numerical radius and p k spectral norm of a respectively for k 1 they are the classical spectral radius numerical radius and spectral norm of a in this note we study the matrices for which p k a rp k a p k a a p k or rp k a a p k moreover the norm properties of rp k and p k are investigated

further inequalities involving the khatri rao product abstract styan g p h styan hadamard products and multivariate statistical analysis linear algebra and its appl 6 1973 217 240 established an inequality involving the hadamard product using statistical reasoning in the context of multivariate analysis in this paper the inequality is extended to involve the khatri rao product in the non negative definite matrix case and in the non singular hermitian matrix case the equality conditions for these extensions are given also established are counterpart inequalities in the positive definite matrix case

necessary and sufficient conditions for multivariate majorization abstract let x and y be m n matrices whose elements are in k a real or complex field we obtain necessary and sufficient conditions for the existence of a matrix a belonging to the convex hull of a certain subgroup of the general linear group gl n k such that x ya which unite and generalize several known results concerning majorization

on permutations convex hulls and normal operators abstract a spectral characterization is obtained for those normal operators which belong to the convex hull of the unitary orbit of a given normal operator on a finite dimensional space this is used to prove the following if a and b are normal operators on an n dimensional complex hilbert space h with eigenvalues given by 1 n and 1 n respectively and if a b is also normal then a b max s n diag k k for any unitarily invariant norm on l h

the relationship between hadamard and conventional multiplication for positive definite matrices abstract it is known that if a is positive definite hermitian then a a 1 i in the positive semidefinite ordering our principal new result is a converse to this inequality under certain weak regularity assumptions about a function f on the positive definite matrices a f a af a for all positive definite a if and only if f a is a positive multiple of a 1 in addition to the inequality a a 1 i it is known that a a 1 t i and stronger that min a b min ab t for a b positive definite hermitian we also show that min a b min ab and note that min ab and min ab t can be quite different for a b positive definite hermitian we utilize a simple technique for dealing with the hadamard product which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under hadamard multiplication and of the inequalities mentioned

some new connections between matrix products for partitioned and non partitioned matrices this paper is concerned with a long list of matrix products for partitioned and non partitioned matrices including kronecker khatri rao of first kind hadamard tracy singh khatri rao block kronecker block hadamard and box products sums our contribution is to gather the most useful connections of the matrix products from various sources and present them in one place several new attractive connections are also added to this collection we give as an example two applications in order to show that these connections play a central role in many applications these applications are a block diagonal least squares problem and a generalization of matrix inequalities involving khatri rao products of positive definite matrices

interpolation in multidimensions a convenient finite dimensional matrix representation of the partial differential operators and some applications an extension to multidimensions of a generalized notion of lagrangian interpolation is used to introduce finite dimensional matrix representations of the partial differential operators this makes it possible to extend to a multidimensional environment various results which were obtained in the past by exploiting such a representation in a one dimensional context some such applications are outlined the construction of remarkable matrices convenient techniques to solve numerically eigenvalue problems and differential equations in multidimensions and the manufacture of solvable many body dynamical systems in multidimensional spaces some instances of such systems are exhibited

imaging quasi layered conductive structures by simple processing of transient electromagnetic data an imaged conductivity section of a layered earth can be obtained by simple transformation of step response electromagnetic data measured in the quasi static zone this method of data transformation is presented as an alternative to conventional apparent conductivity transformations at each delay time the variation of the step response as a function of geometry transmitter and receiver location is transformed to an equivalent reference depth h which can be related to the depth of electromagnetic field diffusion the behavior of h as a function of delay time is nearly independent of the source receiver geometry the slowness dt dh divided by the magnetic permeability is almost exactly proportional to the cumulative conductance measured from the surface down to a depth h thus we can estimate an apparent conductivity which we call the imaged conductivity at depth to be d2t 0dh2 the cost of this transformation is a fraction of the cost of conventional data inversion and it does not require an a

a note on the arithmetic geometric mean inequality for matrices abstract we give a simple proof of the inequality aa x xbb 2 a ab where a b and x are arbitrary n n matrices and is any unitarily invariant norm for the schatten p norms p with 1 p aa x xbb p 2 a xb p if and only if aa x xbb

on majorization and schur products abstract suppose a d1 dm are n n matrices where a is self adjoint and let x m k 1 d k ad k it is shown that if d k d k d k d k i then the spectrum of x is majorized by the spectrum of a in general without assuming any condition on d1 dm a result is obtained in terms of weak majorization if each dk is a diagonal matrix then x is equal to the schur entrywise product of a with a positive semidefinite matrix thus the results are applicable to spectra of schur products of positive semidefinite matrices if a b are self adjoint with b positive semidefinite and if bii 1 for each i it follows that the spectrum of the schur product of a and b is majorized by that of a a stronger version of a conjecture due to marshall and olkin is also proved

occam s inversion a practical algorithm for generating smooth models from electromagnetic sounding data the inversion of electromagnetic sounding data does not yield a unique solution but inevitably a single model to interpret the observations is sought we recommend that this model be as simple or smooth as possible in order to reduce the temptation to overinterpret the data and to eliminate arbitrary discontinuities in simple layered models to obtain smooth models the nonlinear forward problem is linearized about a starting model in the usual way but it is then solved explicitly for the desired model rather than for a model correction by parameterizing the model in terms of its first or second derivative with depth the minimum norm solution yields the smoothest possible model rather than fitting the experimental data as well as possible which maximizes the roughness of the model the smoothest model which fits the data to within an expected tolerance is sought a practical scheme is developed which optimizes the step size at each iteration and retains the computational efficiency of layered models resulting in a stable and rapidly convergent algorithm the inversion of both magnetotelluric and schlumberger sounding field data and a joint magnetotelluric resistivity inversion demonstrate the method and show it to have practical application

schur complements and matrix inequalities in the lowner ordering abstract the purpose of this paper is to present a matrix inequality on the kronecker product that unifies the proofs of many existing matrix inequalities in the lowner partial ordering on the sum ordinary product and hadamard schur product schur complements serve as the basic tool

log majorization and complementary golden thompson type inequalities abstract we obtain a log majorization result for power means of positive semidefinite matrices this implies matrix norm inequalities for unitarily invariant norms which are considered as complementary to the golden thompson one other log majorization results are also obtained we give logarithmic trace inequalities and determinant inequalities as applications of our log majorizations

inversion of horizontal loop electromagnetic frequency soundings a procedure has been developed to perform automatic inversion of hlem frequency soundings by the method of backus and gilbert bg using a linearized iterative scheme a layered conductivity is constructed satisfying the available soundings to an accuracy consistent with the observational uncertainties subsequently in order to investigate the resolving power of the data and hence hopefully to determine the broad scale features which are common to all conductivity models which satisfy the data bg average conductivities are computed for the constructed model at a number of depths in examples with both synthetic data and field data these averages proved to be identical except for random error to the corresponding average conductivities computed for a number of different models satisfying the same data the additional models for this analysis were generated by maximizing or minimizing the conductivities of individual layers in the original constructed model using linear programming techniques on the

inequalities for the schatten p norm iv we present some inequalities for the schattenp norm of operators on a hilbert space it is shown among other things that ifa is an operator such that rea a 0 then for any operatorx ax xa p 2a x p also for any two operatorsa andb a b 2 2 a b 2 2 2 a b 2 2

a note on the hoffman wielandt theorem an elementary proof of a recent generalization of the hoffman wielandt theorem to commuting m tuples of normal matrices by bhatia and bhattacharyya is given in the case m 1 this provides a derivation which though using the same tools avoids some unnecessary calculations and is thus shorter than the original one and that to be found in the standard literature

cauchy schwarz inequalities associated with positive semidefinite matrices abstract using a quasilinear representation for unitarily invariant norms we prove a basic inequality let a l x x m be positive semidefinite where x m m n then x p 2 l p m p for all p 0 and all unitarily invariant norms we show how several inequalities of cauchy schwarz type follow from this bound and obtain a partial analog of our results for l p norms

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p