Let $(A)$ be a complex $(ntimes n)$ matrix and assume that the numerical range of $(A)$ lies in the set of a sector of half angle $(alpha)$ denoted by $(S_{alpha})$. We prove the numerical ranges of the conjugate, inverse and Schur complement of any order of $(A)$ are in the same $(S_{alpha})$.The eigenvalues of some kinds of matrix product and numerical ranges of hadmard product, star-congruen...

An abstract framework for constructing stable decompositions of the spaces corresponding to general symmetric positive definite problems into “local” subspaces and a global “coarse” space is developed. Particular applications of this abstract framework include practically important problems in porous media applications such as: the scalar elliptic (pressure) equation and the stream function for...

In this note, we show that the decomposition group Dec(I) of a zero-dimensional radical ideal I in K[x1, . . . , xn] can be represented as the direct sum of several symmetric groups of polynomials based upon using Gröbner bases. The new method makes a theoretical contribution to discuss the decomposition group of I by using Computer Algebra without considering the complexity. As one application...

We present a family of algorithms for computing symmetric rank-revealing VSV decompositions, based on triangular factorization of the matrix. The VSV decomposition consists of a middle symmetric matrix that reveals the numerical rank in having three blocks with small norm, plus an orthogonalmatrix whose columns span approximations to the numerical range and null space. We show that for semi-de ...

In this work we apply Dykstra’s alternating projection algorithm for minimizing ‖AX − B‖ where ‖ · ‖ is the Frobenius norm and A ∈ Rm×n, B ∈ Rm×n and X ∈ Rn×n are doubly symmetric positive definite matrices with entries within prescribed intervals. We first solve the constrained least-squares matrix problem by using the special structure properties of doubly symmetric matrices, and then use the...

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is symmetric and positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the ...