论文标题
随机符号模式和SIPP的其他应用的正交实现
Orthogonal realizations of random sign patterns and other applications of the SIPP
论文作者
论文摘要
符号模式是一个数组，其中$ \ {+， - ，0 \} $中的条目。如果$ qq^t = i $，矩阵$ q $是行正交的。强大的内部产品特性（SIPP），在[B.A.〜Curtis and B.L.〜着色器中引入，正交矩阵的符号模式以及强大的内部产品属性，线性代数应用。 592：228--259，2020]]是确定符号模式是否允许行正交性的重要工具，因为它可以保证附近的矩阵具有相同的属性，从而使零条件可以扰动到非零条目，同时保留每个非零条目的标志。本文使用SIPP来启动条件研究，在该条件下，随机符号模式允许行较高的可能性正交性。在先前工作的基础上，确定了最小允许正交性的$ 5 \ times n $ nhice零标志模式。建立了标志模式中零条目的条件，以确保具有这种符号模式的任何行正交矩阵具有SIPP。
A sign pattern is an array with entries in $\{+,-,0\}$. A matrix $Q$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A.~Curtis and B.L.~Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228--259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $5\times n$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP.
