Section outline

  • General

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  • 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