Syllabus
1) Exponential of a matrix. Basic properties (similarity of matrices, eigenvectors, exponential of a sum). The relation Tr A = det exp A . Examples (Taylor polynomial, exponential of a commutator)
2) Elementary introduction to Lie algebras. Examples: gl, sl, o, u, su. Isomorphism of vector multiplication resp. commutation in o(3).
3) Nilpotent operators. Basic theorem on their structure and Jordan basis
4) Direct decomposition of a complex vector space according to its spectrum, Jordan theorem. Hamilton Cayley theorem. Exponential of a Jordan cell, applications to systems of linear differential equations with constant coefficients (and special choices of "external forces")
5) Positive and stochastic matrices, interpertation of their spectral radius, applications
6) Dual space, dual bases and operators
7) Duality and scalar product: Adjoint operator, normal operators. Adjoint differential operators and the method of per partes.
8) Spectral decomposition of a normal operator. Examples, Legendre and Hermite polynomials
9) Bilinear and quadratic forms, their diagonalization by a) change of cooordinates (method by "completing the squares") b) Jacobi Sylvester orthogonalization method c) diagonalization by spectral decomposition. Signature
10) Quadratic surfaces and conic sections, their classification (hyperboloids, elipsoids, paraboloids) and basic properties. Projective space.
11) Polar decomposition of an operator
12) Pseudoinverse of a matrix
13) Tensor product of linear spaces, definition, examples, "decomposable" tensors
14) Transformation rules for tensors, covariant and contravariant indices, summation convention
15) Tensor product of tensors, trace of a tensor. Tensors and scalar products, representation of covariant tensors by contravariant ones
16) Symmetric tensors, symmetrization of a (product of) tensor(s)
17) Antisymmetric tensors, antisymmetrization, exterior (Grassmann) algebra. The notion of a k-dimensional volume in n -dimensional vector space. Gramm matrix and the Gramm determinant (for general, non square matrix)
Annotation
This course gives, together with parallel courses on analysis, a basic course of mathematics for physicists. Emphasis is given also to relationship of all these disciplines.
Keywords: selfadjoint operators, quadratic forms, tensors.