We call a matrix triadic if it has no more than two nonzero off-diagonal elements in any column. A symmetric tridiagonal matrix is a special case. In this paper we consider LXLT factorizations of symmetric triadic matrices, where L is unit lower triangular and X is diagonal, block diagonal with 1×1 and 2×2 blocks, or the identity with L lower triangular. We prove that with diagonal pivoting, th...

Motivated by recent interest in group-symmetry in the area of semidefinite programming, we propose a numerical method for finding a finest simultaneous block-diagonalization of a finite number of symmetric matrices, or equivalently the irreducible decomposition of the matrix ∗-algebra generated by symmetric matrices. The method does not require any algebraic structure to be known in advance, wh...

We study block-diagonal preconditioners and an efficient variant of constraint preconditioners for general two-by-two block linear systems with zero (2,2)-block. We derive block-diagonal preconditioners from a splitting of the (1,1)-block of the matrix. From the resulting preconditioned system we derive a smaller, so-called related system that yields the solution of the original problem. Solvin...

Preconditioners based on various multilevel extensions of two-level piecewise linear finite element methods lead to iterative methods which have an optimal order computational complexity with respect to the size (or discretization parameter) of the system. The methods can be in block matrix factorized form, recursively extended via certain matrix polynomial approximations of the arising Schur c...

We show that the growth-factor bound in the Bunch–Kaufman factorization method is essentially tight. The method factors a symmetric matrix A into A 1⁄4 PTLDLTP, where P is a permutation matrix, L is lower triangular, and D is block diagonal with 1-by-1 and 2-by-2 diagonal blocks. The method uses one of several partial pivoting rules that ensure bounded in the elements of the reduced matrix and ...

where N is written in block diagonal form with 2 × 2 matrices appearing along the diagonal, and the mj are real and positive. Moreover, detU = e , where −π < θ ≤ π, is uniquely determined. N is called the real normal form of a non-singular antisymmetric matrix [1–3]. If M is a complex [or real] singular antisymmetric d× d matrix of rank 2n (where d is either even or odd and d > 2n), then there ...

The input autocorrelation matrix for a third order (cubic) Volterra adaptive filter for general colored Gaussian input processes is analyzed to determine how to best formulate a computationally efftcient fast adaptive algorithm. When the input signal samples are ordered properly within the input data vector, the autocorrelation matrix of the cubic filter inherits a block diagonal structure, wit...

We present a fast and numerically stable algorithm for computing the eigendecom-position of a symmetric block diagonal plus semiseparable matrix. We report numerical experiments that indicate that our algorithm is signiicantly faster than the standard method which treats the given matrix as a general symmetric dense matrix.

Face recognition has been widely studied due to its importance in various applications. However, the case that both training images and testing images are corrupted is not well addressed. Motivated by the success of low-rank matrix recovery, we propose a novel semisupervised low-rank matrix recovery algorithm for robust face recognition. The proposed method can learn robust discriminative repre...