Topic outline

  • Determinants

    Definition, properties with respect to matrix operations - transposition, elementary row operations, matrix product, inverse

    Linearity of the determinant

    Determinant calculation - by Gauss elimination and by Laplace expansion

    Adjoint matrix

    Cramer's rule

    Geometric meaning of the determinant

    Application - number of spanning trees of a graph

    • Polynomials

      Polynomial operations - addition, subtraction, products, division with a remainder

      Fermat's little theorem

      Roots, the fundamental theorem of algebra (without a proof)

      Decomposition into monomials

      Representation of polynomials

      Vandermonde matrix and its regularity

      Lagrange interpolation 

      Applications - secret sharing, fast integer multiplication

      • Eigenvalues and eigenvectors

        Eigenvalues and eigenvectors of linear maps and square matrices

        Properties of eigenvectors (subspaces, linear independence)

        Characteristic polynomial and its coefficients

        Calculation of eigenvalues and eigenvectors

        Application in systems of differential equations

        Cayley-Hamilton theorem

      • Diagonalization

        Similar matrices

        Diagonalization - definition, existence via eigenvectors

        Jordan normal form (without a proof)

        Diagonalization of symmetric and Hermitian matrices 

        • Inner spaces

          Inner product, norm

          Cauchy-Schwarz inequality

          Orthogonality, orthonormal bases 

          Orthogonal projection, Gramm-Schmidt orthonormalization

          Isometry

          Orthogonal complement

          • Positive definite matrices

            Gram matrix

            Positive definite matrices

            Cholesky factorization

            Other characterizations

            • Bilinear and quadratic forms

              Bilinear and quadratic forms

              Matrices of forms

              Diagonalization of quadratic forms

              Sylvester's law of inertia

              • Applications

                Number of spanning trees - see the section of determinants

                Number of even subgraphs

                Maximum number of lines spanning the same angle