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"Totally nonnegative matrices arise in a remarkable variety of mathematical applications. This book is a comprehensive and self-contained study of the essential theory of totally nonnegative matrices, defined by the nonnegativity of all subdeterminants. It explores methodological background, historical highlights of key ideas, and specialized topics.The book uses classical and ad hoc tools, but a unifying theme is the elementary bidiagonal factorization, which has emerged as the single most important tool for this particular class of matrices. Recent work has shown that bidiagonal factorizations may be viewed in a succinct combinatorial way, leading to many deep insights. Despite slow development, bidiagonal factorizations, along with determinants, now provide the dominant methodology for understanding total nonnegativity. The remainder of the book treats important topics, such as recognition of totally nonnegative or totally positive matrices, variation diminution, spectral properties, determinantal inequalities, Hadamard products, and completion problems associated with totally nonnegative or totally positive matrices. The book also contains sample applications, an up-to-date bibliography, a glossary of all symbols used, an index, and related references"-- "Totally Nonnegative Matrices" is a comprehensive, modern treatment of the titled class of matrices that arise in very many ways. Methodological background is given, and elementary bidiagonal factorization is a featured tool. In addition to historical highlights and sources of interest, some of the major topics include: recognition, variation diminution, spectral structure, determinantal inequalities, Hadamard products, and completion problems. "--

Non-negative matrices. --- Nonnegative matrices --- Matrices --- EB factorization. --- Hadamard core. --- Hadamard multiplication. --- Hadamard powers. --- Hadamard products. --- IITN matrices. --- Jacobi matrices. --- LU factorization. --- MLBC graphs. --- Perron complements. --- Perron-Frobenius theory. --- TN completions. --- TN linear transformations. --- TN matrices. --- TN matrix structure. --- TN matrix. --- TN perturbations. --- TN polynomial matrices. --- TP intervals. --- TP matrices. --- TP matrix. --- TP polynomial matrices. --- Vandermonde matrices. --- bidiagonal factorization. --- completions. --- constructions. --- core matrix theory. --- detemrinants. --- determinantal identities. --- determinantal inequalities. --- direct summation. --- eigenvalues. --- eigenvectors. --- elementary linear algebra. --- extensions. --- line insertion. --- linear transformations. --- matrix completion problems. --- matrix theory. --- nonnegativity. --- numerical analysis. --- partial TN matrices. --- planar diagrams. --- positive minors. --- positive semidefinite matrices. --- positivity. --- powers. --- principal minors. --- rank deficiency. --- rank deficient submatrices. --- recognition. --- retractions. --- roots. --- sign variation diminution. --- spectral properties. --- statistics. --- subdeterminants. --- subdirect sums. --- total positivity. --- totally nonnegative matrices. --- totally positive matrices. --- triangular factorization. --- variation diminution. --- vectors.

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