This is an updated translation of a book published in Czech by the SNTL - Publishers of Technical Literature in 1981. In developing this book, it was found reasonable to consider special matrices in general sense and also to include some more or less auxiliary topics that made it possible to present some facts or processes more demonstratively. An example is the graph theory. Chapter 1 contains the definitions of basic concepts of the theory of matrices, and fundamental theorems. The Schur complement is defined here in full generality and using its properties we prove the theorem on the factorization of a partitioned matrix into the product of a lower block triangular matrix with identity diagonal blocks, a block diagonal matrix, and an upper block triangular matrix with identity diagonal blocks. The theorem on the Jordan normal form of a matrix is gi¥en without proof. Chapter 2 is concerned with symmetric and Hermitian matrices. We prove Schur's theorem and, using it, we establish the fundamental theorem describing the factorization of symmetric or Hermitian matrices. Further, the properties of positive definite and positive semidefinite matrices are studied. In the conclusion, Sylvester's law of inertia of quadratic forms and theorems on the singular value decomposition and polar decomposition are proved. Chapter 3 treats the mutual connections between graphs and matrices.
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
Research
Illustrationen
Maße
Höhe: 24 cm
Breite: 16 cm
Dicke: 20 mm
Gewicht
ISBN-13
978-90-247-2957-9 (9789024729579)
DOI
10.1007/978-94-009-4335-3
Schweitzer Klassifikation
1. Basic Concepts of the Theory of Matrices.- Matrices.- Determinants.- Nonsingular matrices. Inverse matrices.- Schur complement. Factorization.- Vector spaces. Rank.- Eigenvectors, eigenvalues. Characteristic polynomial.- Similarity. Jordan normal form.- Exercises.- 2. Symmetric Matrices. Positive Definite and Semidefinite Matrices.- Euclidean and unitary spaces.- Symmetric and Hermitian matrices.- Orthogonal and unitary matrices.- Gram-Schmidt orthonormalization. Schur's theorem.- Positive definite and positive semidefinite matrices.- Sylvester's law of inertia.- Singular value decomposition.- Exercises.- 3. Graphs and Matrices.- Digraphs.- Digraph of a matrix.- Undirected graphs. Trees.- Bigraphs.- Exercises.- 4. Nonnegative Matrices. Stochastic and Doubly Stochastic Matrices.- Nonnegative matrices.- The Perron-Frobenius theorem.- Cyclic matrices.- Stochastic matrices.- Doubly stochastic matrices.- Exercises.- 5. M-Matrices (Matrices of Classes K and K0).- Class K.- Class K0.- Diagonally dominant matrices.- Monotone matrices.- Class P.- Exercises.- 6. Tensor Product of Matrices. Compound Matrices.- Tensor product.- Compound matrices.- Exercises.- 7. Matrices and Polynomials. Stable matrices.- Characteristic polynomial.- Matrices associated with polynomials.- Bézout matrices.- Hankel matrices.- Toeplitz and Löwner matrices.- Stable matrices.- Exercises.- 8. Band Matrices.- Band matrices and graphs.- Eigenvalues and eigenvectors of tridiagonal matrices.- Exercises.- 9. Norms and Their Use for Estimation of Eigenvalues.- Norms.- Measure of nonsingularity. Dual norms.- Bounds for eigenvalues.- Exercises.- 10. Direct Methods for Solving Linear Systems.- Nonsingular case.- General case.- Exercises.- 11. Iterative Methods for Solving Linear Systems.- The Jacobi method.- The Gauss-Seidel method.- The SOR method.- Exercises.- 12. Matrix Inversion.- Inversion of special matrices.- The pseudoinverse.- Exercises.- 13. Numerical Methods for Computing Eigenvalues of Matrices.- Computation of selected eigenvalues.- Computation of all the eigenvalues.- Exercises.- 14. Sparse matrices.- Storing. Elimination ordering.- Envelopes. Profile.- Exercises.
