Osnove matrične analize

Abstract

Uvodoma predstavimo matrični račun, sisteme linearnih enačb in determinanto. Nato spoznamo vektorski prostor kot algebrsko strukturo, predstavitev vektorjev z matričnimi stolpci glede na izbrano bazo, pojem vektorskega podprostora ter pomembne podprostore, povezane z matrikami. Nadalje se na kratko posvetimo linearnim preslikavam in njihovi matrični predstavitvi. Analiza značilnih podprostorov, ki so prirejeni matriki, omogoča obravnavo določenih lastnosti ustreznih linearnih preslikav. Vektorski prostor dodatno opremimo še s skalarnim produktom, kar omogoča vpeljavo pojma ortogonalnosti, ta pa pripelje do učinkovite optimizacijske metode, metode najmanjših kvadratov, ki je v inženirski praksi zelo pogosta in uporabna. Obravnavamo osrednji problem linearne algebre oziroma matrične analize, problem lastnih vrednosti. S tem je povezana diagonalizacija matrike, Jordanova normalna oblika in unitarna podobnost trikotni matriki. Slednja na enostaven način omogoči obravnavo hermitskih in simetričnih matrik, ki imajo v inženirski uporabi posebno mesto. Na koncu nanizamo še nekaj primerov uporabe teorije iz prejšnjih poglavij, ki se nanašajo na spektralne lastnosti matrik. Posebej izpostavimo razcep s singularnimi vrednostmi, ki ima zelo široke možnosti uporabe. Učbenik zaključimo s posplošenimi inverzi matrik.

Keywords

matrični račun;matrike;determinante;sistemi linearnih enačb;vektorski prostor;vektorski podprostor;linearna algebra;lastna vrednost;Jordanova normalna oblika;diagonalizacija matrike;posplošitveni inverzi matrik;učbeniki;

Data

Language:	Slovenian
Year of publishing:	2024
Typology:	2.03 - Reviewed University, Higher Education or Higher Vocational Education Textbook
Organization:	UM FERI - Faculty of Electrical Engineering and Computer Science
Publisher:	Univerza v Mariboru, Univerzitetna založba
UDC:	004.422.632:004.422.632(0.034.2)
COBISS:	212044803
ISBN:	978-961-286-911-3
Views:	0
Downloads:	6
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Other data

Secondary language:	English
Secondary title:	Fundamentals of Matrix Analysis
Secondary abstract:	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.
Secondary keywords:	matrix;determinant;system of linear equations;vector space;inner product;norm;eigenvector;eigenvalue;diagonalization;Jordan normal form;singular value decomposition;generalized inverse;
Type (COBISS):	Higher education textbook
Pages:	1 spletni vir (datoteka PDF (II, 148, [4] str.))
DOI:	10.18690/um.feri.7.2024
ID:	25350589