Let A and B be n × n real matrices with A symmetric and B skewsymmetric. Obviously, every simultaneously neutral subspace for the pair (A,B) is neutral for each Hermitian matrix X of the form X = μA + iλB, where μ and λ are arbitrary real numbers. It is well-known that the dimension of each neutral subspace of X is at most In+(X) + In0(X), and similarly, the dimension of each neutral subspace o...

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 superand subdiagonal elements. The corresponding elements of the superand subdiagonal will have the same absolute val...

By applying the generalized singular value decomposition and the canonical correlation decomposition simultaneously, we derive an analytical expression of the optimal approximate solution b X, which is both a least-squares symmetric orthogonal anti-symmetric solution of the matrix equation A XA = B and a best approximation to a given matrix X∗. Moreover, a numerical algorithm for finding this o...

in this paper, we characterize multiresolution analysis(mra) parseval frame multiwavelets in l^2(r^d) with matrix dilations of the form (d f )(x) = sqrt{2}f (ax), where a is an arbitrary expanding dtimes d matrix with integer coefficients, such that |deta| =2. we study a class of generalized low pass matrix filters that allow us to define (and construct) the subclass of mra tight frame multiwav...

The pseudodeterminant pdet(M) of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If ∂ is a symmetric or skewsymmetric matrix then pdet(∂∂t) = pdet(∂)2. Whenever ∂ is the kth boundary map of a self-dual CWcomplex X, this linear-algebraic identity implies that the torsion-weighted generating function for cel...

Some probability distributions (e.g., Gaussian) are symmetric, some (e.g., lognormal) are non-symmetric (skewed). How can we gauge the skeweness? For symmetric distributions, the third central moment C3 def = E[(x − E(x))] is equal to 0; thus, this moment is used to characterize skewness. This moment is usually estimated, based on the observed (sample) values x1, . . . , xn, as C3 = 1 n · n ∑ i...

A square matrix is called line-sum-symmetric if the sum of elements in each of its rows equals the sum of elements in the corresponding column. Let A be an n x n nonnegative matrix and let X and Y be n x n diagonal matrices having positive diagonal elements. Then the matrices XA, XAXand XA Yare called a row-scaling, a similarity-scaling and an equivalence-scaling of A. The purpose of this paper...

We say that X = [xij ]i,j=1 is symmetric centrosymmetric if xij = xji and xn−j+1,n−i+1, 1 ≤ i, j ≤ n. In this paper we present an efficient algorithm for minimizing ‖AXB + CY D − E‖ where ‖ · ‖ is the Frobenius norm, A ∈ Rt×n, B ∈ Rn×s, C ∈ Rt×m, D ∈ Rm×s, E ∈ Rt×s and X ∈ Rn×n is symmetric centrosymmetric with a specified central submatrix [xij ]r≤i,j≤n−r, Y ∈ Rm×m is symmetric with a specifie...

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDXT of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(εκ(X)), where ε is the machine precision and κ(X) ≡ ‖X‖2 · ‖X−1‖2 is the spectral condition number of X . The eigenvectors are a...