Osnove matrične analize

Fundamentals of Matrix Analysis. In the introduction, we present matrix calculus, systems of linear equations and the determinant. Next, we explore the vector space as an algebraic structure, representing vectors with matrix columns based on a chosen basis, the concept of a vector subspace, and important subspaces related to matrices. We then briefly focus on linear transformations and their matrix representation. Analyzing characteristic subspaces associated with a matrix allows us to examine certain properties of the corresponding linear transformations. We further equip the vector space with an inner product, which introduces the concept of orthogonality, leading to an effective optimization method, the least squares method, which is very common and useful in engineering practice. We address the central problem of linear algebra or matrix analysis, the eigenvalue problem. This includes matrix diagonalization, Jordan normal form and unitary similarity to a triangular matrix, which facilitates the treatment of Hermitian and symmetric matrices, which hold a special place in engineering applications. Finally, we list some examples of applying the theory from previous chapters, relating to the spectral properties of matrices. We particularly highlight the singular value decomposition, which has very broad applications. We close the textbook with generalized inverses of matrices.