Special matrices and their applications in numerical mathematics
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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.- Bezout matrices.- Hankel matrices.- Toeplitz and Lowner 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.