Normal Matrices Research Articles

Some Results on Certain Symmetric Circulant Matrices

A direct method for nding the inverse of a class of symmetric circulant matrices is given in [4]. In this paper a method of nding the Moore-Penrose inverse for a class of singular circulant matrices is presented and the spectral norm and spectral radius are calculated. Finally the spectral norm and spectral radius for symmetric circulant matrices with binomial coecients are derived.

Coburn's lemma and the finite section method for random Jacobi operators

We study the spectra and pseudospectra of semi-infinite and bi-infinite tridiagonal random matrices and their finite principal submatrices, in the case where each of the three diagonals varies over a separate compact set, say U , V , W ⊂ C . Such matrices are sometimes termed stochastic Toeplitz matrices A + in the semi-infinite case and stochastic Laurent matrices A in the bi-infinite case. Their spectra, Σ = spec A and Σ + = spec A + , are independent of A and A + as long as A and A + are pseudoergodic (in the sense of Davies (2001) [20] ), which holds almost surely in the random case. This was shown in Davies (2001) [20] for A ; that the same holds for A + is one main result of this paper. Although the computation of Σ and Σ + in terms of U , V and W is intrinsically difficult, we give upper and lower spectral bounds, and we explicitly compute a set G that fills the gap between Σ and Σ + in the sense that Σ ∪ G = Σ + . We also show that the invertibility of one (and hence all) operators A + implies the invertibility – and uniform boundedness of the inverses – of all finite tridiagonal square matrices with diagonals varying over U , V and W . This implies that the so-called finite section method for the approximate solution of a system A + x = b is applicable as soon as A + is invertible, and that the finite section method for estimating the spectrum of A + does not suffer from spectral pollution. Both results illustrate that tridiagonal stochastic Toeplitz operators share important properties of (classical) Toeplitz operators. Indeed, one of our main tools is a new stochastic version of the Coburn lemma for classical Toeplitz operators, saying that a stochastic tridiagonal Toeplitz operator, if Fredholm, is always injective or surjective. In the final part of the paper we bound and compare the norms, and the norms of inverses, of bi-infinite, semi-infinite and finite tridiagonal matrices over U , V and W . This, in particular, allows the study of the resolvent norms, and hence the pseudospectra, of these operators and matrices.

Sharp MSE Bounds for Proximal Denoising

Denoising has to do with estimating a signal $$\mathbf {x}_0$$x0 from its noisy observations $$\mathbf {y}=\mathbf {x}_0+\mathbf {z}$$y=x0+z. In this paper, we focus on the structured denoising where the signal $$\mathbf {x}_0$$x0 possesses a certain structure and $$\mathbf {z}$$z has independent normally distributed entries with mean zero and variance $$\sigma ^2$$ź2. We employ a structure-inducing convex function $$f(\cdot )$$f(·) and solve $$\min _\mathbf {x}\{\frac{1}{2}\Vert \mathbf {y}-\mathbf {x}\Vert _2^2+\sigma {\lambda }f(\mathbf {x})\}$$minx{12źy-xź22+źźf(x)} to estimate $$\mathbf {x}_0$$x0, for some $$\lambda >0$$ź>0. Common choices for $$f(\cdot )$$f(·) include the $$\ell _1$$l1 norm for sparse vectors, the $$\ell _1-\ell _2$$l1-l2 norm for block-sparse signals and the nuclear norm for low-rank matrices. The metric we use to evaluate the performance of an estimate $$\mathbf {x}^*$$xź is the normalized mean-squared error $$\text {NMSE}(\sigma )=\frac{{\mathbb {E}}\Vert \mathbf {x}^*-\mathbf {x}_0\Vert _2^2}{\sigma ^2}$$NMSE(ź)=Eźxź-x0ź22ź2. We show that NMSE is maximized as $$\sigma \rightarrow 0$$źź0 and we find the exact worst-case NMSE, which has a simple geometric interpretation: the mean-squared distance of a standard normal vector to the $${\lambda }$$ź-scaled subdifferential $${\lambda }\partial f(\mathbf {x}_0)$$źźf(x0). When $${\lambda }$$ź is optimally tuned to minimize the worst-case NMSE, our results can be related to the constrained denoising problem $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-xź2}. The paper also connects these results to the generalized LASSO problem, in which one solves $$\min _{f(\mathbf {x})\le f(\mathbf {x}_0)}\{\Vert \mathbf {y}-{\mathbf {A}}\mathbf {x}\Vert _2\}$$minf(x)≤f(x0){źy-Axź2} to estimate $$\mathbf {x}_0$$x0 from noisy linear observations $$\mathbf {y}={\mathbf {A}}\mathbf {x}_0+\mathbf {z}$$y=Ax0+z. We show that certain properties of the LASSO problem are closely related to the denoising problem. In particular, we characterize the normalized LASSO cost and show that it exhibits a phase transition as a function of number of observations. We also provide an order-optimal bound for the LASSO error in terms of the mean-squared distance. Our results are significant in two ways. First, we find a simple formula for the performance of a general convex estimator. Secondly, we establish a connection between the denoising and linear inverse problems.

A Rank-One Tensor Updating Algorithm for Tensor Completion

In this letter, we propose a rank-one tensor updating algorithm for solving tensor completion problems. Unlike the existing methods which penalize the tensor by using the sum of nuclear norms of unfolding matrices, our optimization model directly employs the tensor nuclear norm which is studied recently. Under the framework of the conditional gradient method, we show that at each iteration, solving the proposed model amounts to computing the tensor spectral norm and the related rank-one tensor. Because the problem of finding the related rank-one tensor is NP-hard, we propose a subroutine to solve it approximately, which is of low computational complexity. Experimental results on real datasets show that our algorithm is efficient and effective.

On the harmonic and hyperharmonic Fibonacci numbers

In this paper, we study the theory of the harmonic and the hyperharmonic Fibonacci numbers. Also, we get some combinatoric identities like as harmonic and hyperharmonic numbers and we obtain some useful formulas for $\mathbb{F}_{n}$, which is finite sums of reciprocals of Fibonacci numbers. We obtain spectral and Euclidean norms of circulant matrices involving harmonic and hyperharmonic Fibonacci numbers.

Convergence analysis of the MCGNR algorithm for the least squares solution group of discrete-time periodic coupled matrix equations

The conjugate gradient normal equations residual minimizing (CGNR) algorithm is a popular tool for solving large nonsymmetric linear systems. In this study, we propose the matrix form of the CGNR (MCGNR) algorithm to find the least squares solution group of the discrete-time periodic coupled matrix equations [Formula: see text] with periodic matrices. We prove that the MCGNR algorithm converges in a finite number of steps in the absence of round-off errors, that is, this algorithm has the finite termination property. Also we show that the norms of the residual matrices of the MCGNR algorithm decrease monotonically during its iteration, that is, this algorithm has the residual reducing property. To show the efficiency of the MCGNR algorithm, two numerical examples are presented.

Embedded space–time codes for multiple‐input–multiple‐output systems in Rayleigh fading

In this study, novel designs of embedded space–time multiplexing codes are presented. The space–time codes are constructed initially based on the decomposition of normal matrices that produce diagonal structures. The eigenvalue decomposition of circulant matrices, a type of normal matrices, can generate these codes. A key constraint in the design is that only the diagonal matrices are function of data. Since the components of the diagonal matrix are linear combinations of the data, the codes are called embedded. These codes provide an embedded diversity, full rate and enable multi-user applications with reduced interference. Later constructions in Galois fields are used to reduce the constellations expansion introduced by the dependency of the eigenvalues of the diagonal matrix to the data. The pairwise error probability for the codes is derived. Performance of the codes based on simulation and theoretical analysis, are compared with the conventional spatial-multiplexing codes and other complex orthogonal designs. Simulation results demonstrate the advantage of the proposed codes. Simulation has been done for Rayleigh-fading channels with known channel-state information.

Erratum to: “On an extremal property of normal matrices” [Mathematical Notes 97 (1), 67–73 (2015)

Erratum to: “On an extremal property of normal matrices” [Mathematical Notes 97 (1), 67–73 (2015)

On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers

In this paper, we study norms of circulant and $r-$circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms.

Abstract 5089: Desmoplastic stroma affects growth and invasion of progressively mutated human pancreatic cancer cells in vitro

Abstract Desmoplastic (desmo) stroma is a prognostic marker for pancreatic cancer, yet mechanisms behind its tumorigenic effects remain poorly understood. Tumor stroma, known to act permissive to naïve cells, may exacerbate tumor progression in mutated cells, whereas normal stroma acts restrictive to even aggressively mutated cells. The mechanistic details of these responses remain unclear. Previous studies suggested that a 3D microenvironment (i.e. Matrigel) is necessary for the manifestation of the metastatic phenotype in vitro in aggressively K-Ras mutated pancreatic cancer cells. Here we describe a new 3D in vitro model to study how human pancreatic stromal extracellular matrices (ECMs) interact with pancreatic epithelial cells, how these interactions activate specific integrins and downstream signaling pathways such as PI3K and ERK1/2, and how these signaling pathways may synergize with intrinsic mutational cues to enhance progression of pancreatic cancer. Our model is comprised of physiologically relevant desmo and normal 3D cell-derived matrices as well as polyacrylamide gel-based bioengineered ECMs. The influences of cell-derived and bioengineered ECMs were tested using a series of isogenic pancreatic ductal epithelial cells (PDEC) displaying different aggressive behavior. In preliminary studies, we established that physiological desmo ECMs are tumor permissive (i.e. induce growth and/or invasion) for all PDEC cells (especially kRas mutated cells) when compared with matched restrictive/normal ECMs. In addition, K-Ras cells cultured on stromal activated (i.e. desmo) myofibroblast-derived 3D ECMs resulted in enhanced ERK1/2 activity, similar to what had been previously observed with Matrigel based system. In order to discern distinct cellular and signal responses mediated by cell-derived ECM in the various PDEC cells, we independently modulated microenvironmental cues such as stiffness, topography, and biochemical composition in the bioengineered ECMs. Our initial data suggests that the various PDEC cells differentially respond to alterations in microenvironmental cues. We anticipate that our studies will lead to better understanding of ECM-PDEC interactions in more detail, and help in identification of novel biomarker for early detection or prevention of pancreatic cancer. Citation Format: Ruchi Malik, Tiffany Luong, Seda Karamil, Peter Lelkes, Edna Cukierman. Desmoplastic stroma affects growth and invasion of progressively mutated human pancreatic cancer cells in vitro. [abstract]. In: Proceedings of the 106th Annual Meeting of the American Association for Cancer Research; 2015 Apr 18-22; Philadelphia, PA. Philadelphia (PA): AACR; Cancer Res 2015;75(15 Suppl):Abstract nr 5089. doi:10.1158/1538-7445.AM2015-5089

Eigenvalue Problem, Spin Systems, Lie Groups, and Parameter Dependence

AbstractWe study square matrices F(α) over ℂ with α∈ℝ, where the eigenvalues depend on the parameter α but not the eigenvectors, and vice versa, where the eigenvectors depend on the parameter α but not the eigenvalues. We derive sufficient conditions for such properties. Applications to Lie groups and spin systems are provided. Both normal and nonnormal matrices are investigated.

A normalizing isospectral flow on complex Hessenberg matrices

We study an isospectral flow (Lax flow) that provides an explicit deformation from upper Hessenberg complex matrices to normal matrices, extending to the complex case and to the case of normal matrices the results of [2]. The Lax flow is given bydAdt=[[A†,A]du,A], where brackets indicate the usual matrix commutator, [A,B]:=AB−BA, A† is the conjugate transpose of A and the matrix [A†,A]du is the matrix equal to [A†,A] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A0 is an upper Hessenberg matrix with simple spectrum, then limt→+∞⁡A(t) exists and it is a normal upper Hessenberg matrix isospectral to A0 and if the spectrum of A0 is contained in a line in the complex plane, then the ω-limit set is actually a tridiagonal normal matrix. Furthermore, we show that this flow is also the solution of an infinite time horizon optimal control problem and we prove that it can be used to construct even dimensional real skew-symmetric tridiagonal matrices with given simple spectrum, and with given signs pattern for the codiagonal elements.

The Schur–Horn problem for normal operators

We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than 3 . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I∞, II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

Representation of real normal (T + H) matrices in the case where the skew-symmetric parts of both summands are skew-circulant matrices

Representation of real normal (T + H) matrices in the case where the skew-symmetric parts of both summands are skew-circulant matrices

A Conjecture on the Hausdorff Dimension of Attractors of Real Self-Projective Iterated Function Systems

In a recent work, we proved that under natural conditions satisfied in several important examples, the rate of growth 1/sA of norms of matrices in a semigroup (respectively ) dictates the Hausdorff dimension of the attractor RA of the corresponding semigroups of projective transformations on (respectively ). In the present work, we begin a study of the higher-dimensional case. In particular, we introduce a certain family of semigroups , and we study numerically concrete cases for n = 3 and n = 4. Our results suggest that for n ≥ 3, (n − 1)sA/n is a lower bound for the Hausdorff dimension of RA.

Orthogonal sets of normal or conjugate-normal matrices

We point out several implications of a result of Djoković (1971) [8] concerning orthogonality of normal matrices that satisfy a certain condition on the eigenvalues of their sum. We obtain an analogous result in the setting of conjugate-normal matrices.

Matrices commuting with a given normal tropical matrix

Consider the space Mnnor of square normal matrices X=(xij) over R∪{−∞}, i.e., −∞≤xij≤0 and xii=0. Endow Mnnor with the tropical sum ⊕ and multiplication ⊙. Fix a real matrix A∈Mnnor and consider the set Ω(A) of matrices in Mnnor which commute with A. We prove that Ω(A) is a finite union of alcoved polytopes; in particular, Ω(A) is a finite union of convex sets. The set ΩA(A) of X such that A⊙X=X⊙A=A is also a finite union of alcoved polytopes. The same is true for the set Ω′(A) of X such that A⊙X=X⊙A=X.A topology is given to Mnnor. Then, the set ΩA(A) is a neighborhood of the identity matrix I. If A is strictly normal, then Ω′(A) is a neighborhood of the zero matrix. In one case, Ω(A) is a neighborhood of A. We give an upper bound for the dimension of Ω′(A). We explore the relationship between the polyhedral complexes spanA, spanX and span(AX), when A and X commute. Two matrices, denoted A̲ and A¯, arise from A, in connection with Ω(A). The geometric meaning of them is given in detail, for one example. We produce examples of matrices which commute, in any dimension.

Universal spectral correlations in ensembles of random normal matrices

We consider non-Gaussian ensembles of random normal matrices with the constraint that the ensembles are invariant under unitary transformations. We show that the level density of eigenvalues exhibits disk or ring structure in the complex plane. We also show that the n-eigenvalue correlation and the spacing distribution are universal and identical to that of complex (Gaussian) Ginibre ensemble. Our results are confirmed by Monte Carlo calculations. We verify the universality for dissipative quantum kicked rotor systems.

The trace norm of r-partite graphs and matrices

The trace norm of graphs has been widely studied under the name graph energy. This note presents bounds on the maximum trace norm of an r-partite graph of order n. The lower bounds come from conference and Hadamard matrices.

Modification of readings along oblique principal meridians to fit regular corneal surfaces

Anterior astigmatic corneal surface elements may have their continuity disturbed by environmental factors. Powers detected along principal meridians define matrices whose natures distinguish the continuity of surface elements. This work aims to adapt the matrix nature of data to fit closely, uniquely and holistically with the matrix nature in a sample of data for smooth astigmatic corneas that have powers along rectangular meridians. Matrices from continuous surfaces approximate matrices with outliers from irregular surfaces explicitly. Only approximations that minimize the Frobenius norm of the residual matrices are selected. Data from smooth astigmatic corneal surfaces, in an augmented sample, may now produce more reliable analyses.