Sign Patterns of Row Orthogonal Matrices
- ISBN-10
- ISBN-13
- 9798662412774
- Category
- Matrices
- Pages
- 139
- Language
- English
- Published
- 2020
- Author
- Bryan A. Curtis
Description
A sign pattern is a matrix with entries coming from the set {1, −1, 0}. The entries of a sign pattern represent positive, negative and 0 entries of a real matrix. The set of real matrices with entries that have the same sign as the corresponding entries of a sign pattern S is denoted Q(S). A sign pattern S allows orthogonality if there exists a row orthogonal matrix in Q(S). New necessary conditions and sufficient conditions for a sign pattern to allow orthogonality are found. A set of linear algebraic conditions, referred to as the strong inner product property, are developed and can be used to construct families of sign patterns that allow orthogonality. These results are used to improve the best-known characterizations of sign patterns that allow orthogonality. Some results are implemented as a computer program for matrices with small dimensions. The necessary conditions that are developed greatly improve the best-known results. New techniques are also used to prove known results. The presentation of the necessary conditions is intended to help highlight the most important combinatorial structure of sign patterns that allow orthogonality. Many new examples of sign patterns that do not allow orthogonality are found. In general, strong properties are derived from the transverse intersection of smooth manifolds to obtain perturbation results that can often be used to answer combinatorial questions. The strong inner product property is a new strong property. Given one row orthogonal matrix satisfying the strong inner product property new sign patterns that allow orthogonality can be found. Let O(m, n, k) denote the set of matrices with spectral norm 1 and singular value 1 with multiplicity k. Many of the results for row orthogonal matrices are generalized to O(m, n, k). This includes the development of a second new strong property, the strong spectral norm property.
Get the book
Similar books
-
矩阵分析
By 约翰逊, Johnson, Horn本书从数学分析的角度论述矩阵分析的经典方法和现代方法,主要内容包括:特征值、特征向量和相似性,酉等价和正规矩阵等。
-
Goethals-Seidel Type Hadamard Matrices
By Jennifer Seberry, Jennifer Seberry WallisGoethals-Seidel Type Hadamard Matrices
-
Construction of Williamson Type Matrices
By Jennifer Seberry, Jennifer Seberry WallisConstruction of Williamson Type Matrices
-
Orthogonal (0,1,-1) - Matrices
By Jennifer Seberry, Jennifer Seberry WallisOrthogonal (0,1,-1) - Matrices
-
A New Construction for Williamson-type Matrices
By Jennifer SeberryA New Construction for Williamson-type Matrices
-
Matrix Methods for Optical Layout
By Gerhard KloosTT64 • High - Fidelity Medical Imaging Displays , Aldo Badano , Michael J. Flynn , and Jerzy Kanicki , Vol . TT63 • Diffractive Optics - Design , Fabrication , and Test , Donald C. O'Shea , Thomas J. Suleski , Alan D. Kathman ...
-
Topics in Matrix Analysis: 卷2
Topics in Matrix Analysis: 卷2
-
Matrix Methods: An Introduction
By Richard BronsonThis new edition of Matrix Methods emphasizes applications to Jordan-canonical forms, differential equations, and least squares.
-
Matrix Computations
By Gene Howard GolubThis new edition includes thoroughly revised chapters on matrix multiplication problems and parallel matrix computations, expanded treatment of CS decomposition, an updated overview of floating point arithmetic, a more accurate rendition of ...
-
Matrix Computations
By Charles F. Van Loan, Gene Howard GolubMatrix Computations