The Theory of Matrices, Volume 1

Volume 1: I. 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 II. 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 III. 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 IV. 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 Bezout 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 V. 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 VI. Equivalent transformations of polynomial matrices. Analytic theory of elementary divisors: 1. Elementary transformations of a polynomial matrix; 2. Canonical form of a $\lambda$-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 VII. 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 VIII. 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 $f(X)=O$; 5. Matrix polynomial equations; 6. The extraction of $m$-the roots of a non-singular matrix; 7. The extraction of $m$-th roots of a singular matrix; 8. The logarithm of a matrix IX. 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 X. 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.