Zeros in positive definite correlation matrices arise frequently in probability and statistics, and are intimately related to the notion of stochastic independence. The question of when zeros (i.e., sparsity) in a positive definite matrix A are preserved in its Cholesky decomposition, and vice versa, was addressed by Paulsen et al. [19] [see Journal of Functional Analysis, 85, 151-178]. In part...

Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive definite Toeplitz matrix. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exp...

Nakajima and Tanaka showed that the algebraic eigenvalue problem occurring in the discrete ordinate and matrix operator methods can be reduced to finding eigenvalues and eigenvectors of the product of two symmetric matrices, one of which is positive definite. Here, we show that the Cholesky decomposition of this positive definite matrix can be used to convert the eigenvalue problem into one inv...

In this paper, some elementary operations on triangular fuzzynumbers (TFNs) are defined. We also define some operations on triangularfuzzy matrices (TFMs) such as trace and triangular fuzzy determinant(TFD). Using elementary operations, some important properties of TFMs arepresented. The concept of adjoints on TFM is discussed and some of theirproperties are. Some special types of TFMs (e.g. pu...

1 Linear algebra 2 1.1 Inner product, norm, distance, and orthogonality . . . . . . . . . 2 1.2 Angle and inequality . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Vector projection . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Basics of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . ....