a matrix $pintextmd{c}^{ntimes n}$ is called a generalized reflection matrix if $p^{h}=p$ and $p^{2}=i$‎. ‎an $ntimes n$‎ ‎complex matrix $a$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $p$ if $a=pap$ ($a=-pap$)‎. ‎in this paper‎, ‎we introduce two iterative methods for solving the pair of matrix equations $axb=c$ and $dxe=f$ over reflexiv...

Rotations are essential transformations in many parts of numerical linear algebra. In this paper, it is shown that there exists a family of matrices unitary with respect to an orthosymmetric scalar product J , that can be decomposed into the product of two J-unitary matrices—a block diagonal matrix and an orthosymmetric block rotation. This decomposition can be used for computing various one-si...

The recent emergence of the discrete fractional Fourier transform (DFRFT) has caused a revived interest in the eigenanalysis of the discrete Fourier transform (DFT) matrix F with the objective of generating orthonormal Hermite-Gaussian-like eigenvectors. The Grünbaum tridiagonal matrix T – which commutes with matrix F – has only one repeated eigenvalue with multiplicity two and simple remaining...

Rotations are essential transformations in many parts of numerical linear algebra. In this paper, it is shown that there exists a family of matrices unitary with respect to an orthosymmetric scalar product J , that can be decomposed into the product of two J-unitary matrices—a block diagonal matrix and an orthosymmetric block rotation. This decomposition can be used for computing various one-si...

We characterize the spectral behavior of a primal Schur-complement-based block diagonal preconditioner for saddle point systems, subject to low-rank modifications. This is motivated by a desire to reduce as much as possible the computational cost of matrix-vector products with the (1,1) block, while keeping the eigenvalues of the preconditioned matrix reasonably clustered. The formulation leads...

A sequence of approximations for the determinant and its logarithm of a complex matrix is derived, along with relative error bounds. The determinant approximations are derived from expansions of det(X) = exp(trace(log(X))), and they apply to non-Hermitian matrices. Examples illustrate that these determinant approximations are efficient for lattice simulations of finite temperature nuclear matte...

It is shown that if the decoherence matrix corresponding to a qubit master equation has a block-diagonal real part, then the evolution is determined by a one-dimensional oscillator equation. Further, when the decoherence matrix itself is block-diagonal, then the necessary and sufficient conditions for completely positive evolution may be formulated in terms of the oscillator Hamiltonian or Lagr...

We present a recursive way to partition hypergraphs which creates and exploits hypergraph geometry and is suitable for many-core parallel architectures. Such partitionings are then used to bring sparse matrices in a recursive Bordered Block Diagonal form (for processor-oblivious parallel LU decomposition) or recursive Separated Block Diagonal form (for cache-oblivious sparse matrix–vector multi...

For the simulation and optimization of large-scale chemical processes, the overall computing time is often dominated by the time needed to solve a large sparse system of linear equations. We describe here a parallel frontal solver which can significantly reduce the wallclock time required to solve these linear equation systems using parallel/vector supercomputers. The algorithm exploits both mu...

In this paper, we propose a PID control design method for decentralized control systems from the inverse linear quadratic problem viewpoint by using the structure of controllers obtained by the ILQ(Inverse Linear Quadratic) design method. First, we show an optimal decentralized PID design method for systems with relative degree no more than 2 and block diagonal decoupling matrix. Then, we deriv...