F. R. Gantmacher

1. Matrices and operations on matrices:
1. Matrices. Basic notation;
2. Addition and multiplication of rectangular matrices;
3. Square matrices;
4. Compound matrices. Minors of the inverse matrix
2. The algorithm of Gauss and some of its applications:
1. Gauss’s elimination method;
2. Mechanical interpretation of Gauss’s algorithm;
3. Sylvester’s determinant identity;
4. The decomposition of a square matrix into triangular factors;
5. The partition of a matrix into blocks. The technique of operating with partitioned matrices. The generalized algorithm of Gauss
3. Linear operators in an n-dimensional vector space:
1. Vector spaces;
2. A linear operator mapping an n-dimensional space into an m-dimensional space;
3. Addition and multiplication of linear operators;
4. Transformation of coordinates;
5. Equivalent matrices. The rank of an operator. Sylvester’s inequality;
6. Linear operators mapping an n-dimensional space into itself;
7. Characteristic values and characteristic vectors of a linear operator;
8. Linear operators of simple structure
4. The characteristic polynomial and the minimal polynomial of a matrix:
1. Addition and multiplication of matrix polynomials;
2. Right and left division of matrix polynomials;
3. The generalized Bézout theorem;
4. The characteristic polynomial of a matrix. The adjoint matrix;
5. The method of Faddeev for the simultaneous computation of the coefficients of the characteristic polynomial and of the adjoint matrix;
6. The minimal polynomial of a matrix
5. Functions of matrices:
1. Definition of a function of a matrix;
2. The Lagrange-Sylvester interpolation polynomial;
3. Other forms of the definition of f(A). The components of the matrix A;
4. Representation of functions of matrices by means of series;
5. Application of a function of a matrix to the integration of a system of linear differential equations with constant coefficients;
6. Stability of motion in the case of a linear system
6. Equivalent transformations of polynomial matrices. Analytic theory of elementary divisors:
1. Elementary transformations of a polynomial matrix;
2. Canonical form of a λ-matrix;
3. Invariant polynomials and elementary divisors of a polynomial matrix;
4. Equivalence of linear binomials;
5. A criterion for similarity of matrices;
6. The normal forms of a matrix;
7. The elementary divisors of the matrix f(A);
8. A general method of constructing the transforming matrix;
9. Another method of constructing a transforming matrix
7. The structure of a linear operator in an n-dimensional space:
1. The minimal polynomial of a vector and a space (with respect to a given linear operator);
2. Decomposition into invariant subspaces with co-prime minimal polynomials;
3. Congruence. Factor space;
4. Decomposition of a space into cyclic invariant subspaces;
5. The normal form of a matrix;
6. Invariant polynomials. Elementary divisors;
7. The Jordan normal form of a matrix;
8. Krylov’s method of transforming the secular equation
9. Matrix equations:
1. The equation AX=XB;
2. The special case A=B. Commuting matrices;
3. The equation AX−XB=C;
4. The scalar equation f(X)=O; Matrix polynomial equations;
5. The extraction of m-th roots of a non-singular matrix;
6. The extraction of m-th roots of a singular matrix;
7. The logarithm of a matrix
10. Linear operators in a unitary space:
1. General considerations;
2. Metrization of a space;
3. Gram’s criterion for linear dependence of vectors;
4. Orthogonal projection;
5. The geometrical meaning of the Gramian and some inequalities;
6. Orthogonalization of a sequence of vectors;
7. Orthonormal bases;
8. The adjoint operator;
9. Normal operators in a unitary space;
10. The spectra of normal, hermitian, and unitary operators;
11. Positive-semidefinite and positive-definite hermitian operators;
12. Polar decomposition of a linear operator in a unitary space. Cayley’s formulas;
13. Linear operators in a euclidean space;
14. Polar decomposition of an operator and the Cayley formulas in a euclidean space;
15. Commuting normal operators
11. Quadratic and hermitian forms:
1. Transformation of the variables in a quadratic form;
2. Reduction of a quadratic form to a sum of squares. The law of inertia;
3. The methods of Lagrange and Jacobi of reducing a quadratic form to a sum of squares;
4. Positive quadratic forms;
5. Reduction of a quadratic form to principal axes;
6. Pencils of quadratic forms;
7. Extremal properties of the characteristic values of a regular pencil of forms;
8. Small oscillations of a system with n degrees of freedom;
9. Hermitian forms;
10. Hankel forms

Bibliography
Index