A divide-and-conquer method for computing approximate eigenvalues and eigenvectors of a block tridiagonal matrix is presented. In contrast to a method described earlier [W. N. Gansterer, R. C. Ward, and R. P. Muller, ACM Trans. Math. Software, 28 (2002), pp. 45–58], the off-diagonal blocks can have arbitrary ranks. It is shown that lower rank approximations of the off-diagonal blocks as well as...

Matrix theory plays an important role in precoding methodology for multiple input multiple output (MIMO) systems. In this paper, an improved block diagonal (BD) precoding scheme is proposed for a MIMO multicast channel with two users, where the unitary precoding matrix is constructed in a block-wise form by joint triangularization decomposition. In order to reduce large signal-to-noise ratios (...

We consider uncertain linear systems where the uncertainties, in addition to being bounded, also satisfy constraints on their phase. In this context, we deene the \phase-sensitive structured singular value" (PS-SSV) of a matrix, and show that suucient (and sometimes necessary) conditions for stability of such uncertain linear systems can be rewritten as conditions involving PS-SSV. We then deri...

Mastronardi and Van Dooren [this journal, 34 (2013) pp. 173–196] recently introduced the block antitriangular (“Batman”) decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated in...

Let A be a matrix with its Moore-Penrose pseudo-inverse † . It is proved that, after re-ordering the columns of , projector P = I − has block-diagonal form, that there permutation Π such T diag ( S 1 2 … k ) further each block i corresponds to cluster are linearly dependent other. clustering algorithm provided allows partition into clusters where in correlate only within same cluster. Some appl...

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. ...

Recently, Lee and Hou (IEEE Signal Process Lett 13: 461-464, 2006) proposed one-dimensional and two-dimensional fast algorithms for block-wise inverse Jacket transforms (BIJTs). Their BIJTs are not real inverse Jacket transforms from mathematical point of view because their inverses do not satisfy the usual condition, i.e., the multiplication of a matrix with its inverse matrix is not equal to ...

The curvature matrix depends on the specific optimisation method and will often be only an estimate. For notational simplicity, the dependence of f̂ on θ is omitted. Setting C to the true Hessian matrix of f would make f̂ the exact secondorder Taylor expansion of the function around θ. However, when f is a nonlinear function, the Hessian can be indefinite, which leads to an ill-conditioned quadra...

In this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring R such that, every regular matrix admits a diagonal power-reduction.

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton polytopes, more recently have been shown give rise toric degenerations of various families varieties. Whenever a matching field gives degeneration, the associated polytope variety coincides with polytope. We combinatorial mutations, which are analogues cluster mutations for polytopes show that property giving de...