Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

Diagonal Matrix Scaling and Linear Programming

Abstract: A positive semidefinite symmetric matrix either has a nontrivial nonnegative zero or can be scaled by a positive diagonal matrix into a doubly quasi-stochastic matrix. This paper describes a simple path-following Newton algorithm of the complexity $O(\sqrt{n} L)$ iterations to either scale an $n \times n$ matrix or give a nontrivial nonnegative zero. The latter problem is well known to be equivalent to linear programming.

Symmetric Matrix Derivatives with Applications

Abstract: Dwyer (1967) provided extensive formulas for matrix derivatives, many of which are for derivatives with respect to symmetric matrices. The results of his article are only for symmetric matrices whose (j, i) element is considered to differ from the (i, j) element even though their scalar values are equal. A simple result is given that extends the use of Dwyer's formulas to symmetric matrices in which the (i, j) element is considered functionally equal to the (j, i) element. Application of the results are illustrated by deriving the multivariate normal information matrix.

The Stable Perturbation of the Drazin Inverse of the Square Matrices

Abstract: For any $n\times n$ complex matrix $A$, let $A^D$ and $A^\pi$ be the Drazin inverse and the spectral projector of $A$, respectively, where $A^\pi=I-AA^D$. When $A$ is singular, an $n\times n$ complex matrix $B$ is said to be a stable perturbation of $A$ if $I-A^\pi-B^\pi$ is nonsingular or, equivalently, if the matrix $B$ satisfies condition (${\cal C}_s$) recently introduced by Castro-Gonzalez, Robles, and Velez-Cerrada [SIAM J. Matrix Anal. Appl., 30 (2008), pp. 882-897]. In the perturbation analysis of the Drazin inverse, the condition of $\Vert B-A\Vert$ being small is usually implicitly assumed in the literature. In this case, the condition of $B$ being a stable perturbation of $A$ is necessary in order to ensure the continuity of the Drazin inverse. In this paper, only under the condition that $B$ is a stable perturbation of $A$, two explicit formulas for the Drazin inverse $B^D$ and the spectral projector $B^\pi$ are provided, respectively, and some upper bounds for $\Vert B^D-A^D\Vert/\Vert A^D\Vert$ and $\Vert B^\pi-A^\pi\Vert$ are derived from these formulas under certain conditions. In the case where $\Vert B-A\Vert$ is small, numerical examples are given indicating the sharpness of these norm upper bounds. Furthermore, a numerical example is provided to illustrate that a perturbation analysis of the Drazin inverse can also be conducted, even if $\Vert B-A\Vert$ is not small.

Apparatus for encoding and decoding of low-density parity-check codes, and method thereof

Abstract: An LDPC code encoding apparatus includes: a code matrix generator for generating and transmitting a parity-check matrix comprising a combination of square matrices having a unique value on each row and column thereof; an encoding means encoding block LDPC codes according to the parity-check matrix received from the code matrix generator; and a codeword selector for puncturing the encoded result of the encoding means to generate an LDPC codeword. The code matrix generator divides an information word to be encoded into block matrices having a predetermined length to generate a vector information word. The encoding means encodes the block LDPC codes using the parity-check matrix divided into the block matrices and a Tanner graph divided into smaller graphs in correspondence to the parity-check matrix.