Syllabus
Inner product spaces:
- norm induced by an inner product
- Pythagoras theorem, Cauchy-Schwarz inequality, triangle inequality
- orthogonal and orthonormal system of vectors, Fourier coefficients, Gram-Schmidt orthogonalization
- orthogonal complement, orthogonal projection
- the least squares method
- orthogonal matrices
Determinants:
- basic properties
- Laplace expansion of a determinant, Cramer's rule
- adjugate matrix
- geometric interpretation of determinants
Eigenvalues and eigenvectors:
- basic properties, characteristic polynomial
- Cayley-Hamilton theorem
- similarity and diagonalization of matrices, spectral decomposition, Jordan normal form
- symmetric matrices and their spectral decomposition
- (optionally) companion matrix, estimation and computation of eigenvalues: Gershgorin discs and power method
Positive semidefinite and positive definite matrices:
- characterization and properties
- methods: recurrence formula, Cholesky decomposition, Gaussian elimination, Sylvester's criterion
- relation to inner products
Bilinear and quadratic forms:
- forms and their matrices, change of a basis
- Sylvester's law of inertia, diagonalization, polar basis
Topics on expansion (optionally):
- eigenvalues of nonnegative matrices
- matrix decompositions: Householder transformation, QR, SVD, Moore-Penrose pseudoinverse of a matrix
Annotation
Continuation of MAI057 - special matrices, determinants, eigenvalues, examples of applications of linear algebra.