This chapter is a positive matrix version of Chapter IV. First we give a complete characterization of all 2 by 2 and 3 by 3 positive block matrices. Then this is used to obtain some standard results for positive Toeplitz matrices and the Levinson algorithm. We also show that there is a ...
Let be a real symmetric block-matrix with A and B positive definte. Then M is positive definte if and only if. An application for this result to the stability of constant coefficient linear system is presented.关键词: Positive definite condition block-matrix linear ordinary differential equatio...
Hence, the matrix on the right-hand side of (6.136) is positive definite as well. The main goal of this section is to prove Theorem 6.23. Lemma 6.18 is used in the proof of Lemma 6.19, and Lemma 6.19 is used in the proof of Lemma 6.20, which in turn, is used in the proof of ...
In the first one, we describe the structure of a n脳n block-matrix by the use of a sequence of free parameters (called here generalized choice sequence). This parametrization can be viewed as an adaptation of some ideas from the classical paper of I.Schur [15]. In the second section ...
Perform Cholesky factorization of a symmetric positive definite block tridiagonal matrix. Solution To perform Cholesky factorization of a symmetric positive definite block tridiagonal matrix, with N square blocks of size NB by NB: Perform Cholesky factorization of the first diagonal block. Repeat ...
For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\\begin{pmatrix} A \\& X\\\ {X^*} \\& B\\end{pmatrix} \\in {\\mathbb{M}}\\_{n+m}^+$, with $A\\in {\\mathbb{M}}\\_n^+$, $B \\in {\\mathbb{M}}\\_m^+.$ Th...
For positive semi-definite block-matrix $M,$ we say that $M$ is P.S.D. and we write $M=\begin{pmatrix} A \& X\\ {X^*} \& B\end{pmatrix} \in {\mathbb{M}}\_{n+m}^+$, with $A\in {\mathbb{M}}\_n^+$, $B \in {\mathbb{M}}\_m^+.$ The focus is on studyi...
Linear systems of the form = , where the matrix is symmetric and positive definite, often arise from the discretization of elliptic partial differential equations. A very successful method for solving these linear systems is the preconditioned conjugate gradient method. In this paper, we study parall...
Positive definite, i.e., for any finite collection (gi) of elements ofGthe matrix with entries\chi(g_{i} g_{j}^{-1})is a Hermitian non-negative definite matrix. Ifg↦Rgis a finite-dimensional matrix representation of a groupG, then its trace\chi(g)=\operatorname{Trace}(R_{g})...
英文: Results linear complementary problem have unique solution when M is generalized positive definite matrix.中文: 结果得到了当M是广义正定矩阵时,线性互补问题存在唯一解。英文: When taking, the pawn goes one square diagonally forward.中文: 当吃子时,兵向前斜着走一步。