Course information

Eigenvalues and eigenvectors of linear transformations and matrices: Definitions and theorems, diagonalization of matrices under similarity, Cayley-Hamilton theorem, minimum polynomial. Applications of diagonalization of matrices: Exponential matrix function, linear differential systems, and discrete dynamical linear systems. Inner product linear spaces: Definition of (real and complex) inner product, Cauchy-Schwarz inequality, and definition of cosine of an angle, orthogonality, and projection of a vector. Gram-Schmidt orthonormalization method, orthonormal basis, orthogonal complement. Linear transformations in Euclidean and unitary spaces: Orthogonal and unitary transformations and matrices, the adjoint transformation, diagonalization under unitary similarity of real symmetric and complex Hermitian matrices, positive and negative definite and semidefinite matrices. Factorizations of matrices: QR factorization, SVD factorization, least squares problem, pseudoinverse matrix. Bilinear and quadratic forms: Definitions and canonical form. Applications in classification of curves and surfaces of second degree. Canonical forms of matrices, Jordan canonical form of matrices, rational canonical form of matrices.