Pseudoeigenvector bases and deflated GMRES for highly nonnormal matrices

Pseudoeigenvalues have been extensively studied for highly nonnormal matrices. This paper focuses on the corresponding pseudoeigenvectors. The properties and uses of pseudoeigenvector bases are investigated. It is shown that pseudoeigenvector bases can be much better conditioned than eigenvector bases. We look at the stability and the varying quality of pseudoeigenvector bases. Then applications are considered including the exponential of a matrix. Several aspects of GMRES convergence are looked at, including why using approximate eigenvectors to deflate eigenvalues can be effective even when there is not a basis of eigenvectors.     

On spectral variation of a nonnormal matrix

In this paper we estimate the difference between the spectrum of A (which is supposed to be known completely) and A+E, in terms of the norms of E and $\text{AA}{}^1\text{? A}{}^1\text{A}$. Also we find a special upper triangular form K = Q^-1AQ which separates the distinct eigenvalues of A, and we estimate the 6Q^-16₂6Q6₂ from the above.     

Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

On double-one matrices and double-zero matrices

We investigate two special classes of matrices over GF(2) with certain interesting properties. These properties can be applied to construct nonsingular matrix pairs efficiently and thus provide a solution to the long-key problems of McEIiece's public-key cryptosystem.     

关于径向矩阵与谱矩阵

§1.引言与记号 设A∈C~(s×n),则称 ‖A‖=‖AX‖/‖X‖ 为A的谱模(谱范数),其中‖X‖表示向量X∈C~(n×1)的Euclid范数。即当X=(x_1,…,x_n)~(?)时,‖X‖=(XX)~1/2=sum from i=1 to n(|X_1|~2)~1/2;‖AX‖为向量AX的Euclid范数。 如众周知,我们有如下结论: 引理 1[1]、设A、B∈C~(n×n),则谱模满足范数的三个条件: 1>.恒正性:‖A‖≥0且‖A‖=0 A=0; 2>.齐次性:若α∈C,则‖αA‖=|α|·‖A‖; 3>.三角不等式:‖A+B‖≤‖A‖+‖B‖。     

Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.     

On copositive matrices

A criterion for copositive matrices is given and for n = 3 the set of all copositive matrices is determined in terms of matrix elements. Copositive matrices are applied to the problem of excluding periodic solutions of certain algebraic differential equations.     

On skew-Hadamard matrices

Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.     

On -Hadamard matrices

In this paper, we use the Sylvester's approach to construct another Hadamard matrix, namely a J_m-Hadamard matrix, from a given one. Consequently, we can generate other 2^m-1 Hadamard matrices from the constructed J_m-Hadamard matrix. Finally, we also discuss the Kronecker product of an Hadamard matrix and a J_m-Hadamard matrix.     

On nonnormal products and a set of A.H. Stone

A variant of Michael's example is given to the following effect: there is a Lindelöf space M of weight ℵ₁, with all Gδ-sets open, such that M×B(ℵ₁) is nonnormal. This answers a question from [K. Alster, On the class of ω₁-metrizable spaces whose product with every paracompact space is paracompact, Topology Appl. 153 (2006) 2508-2517].     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

On tridiagonal conjugate-normal matrices

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.     

On integralEP r matrices

Periodica Mathematica Hungarica - This paper gives a characterization of integralEP r matrices and necessary and sufficient conditions for the generalized inverse of the product of two integralEP r...     

On full-rank interval matrices

For interval matrices, the paper considers the problem of determining whether a matrix has full rank. We propose a full-rank criterion that relies on the search for diagonal dominance as well as criteria based on pseudoinversion of the midpoint matrix and comparison of the midpoint and the radius matrices for the interval matrix under study.