Analysis of Matrices

A tantárgy neve magyarul / Name of the subject in Hungarian: Mátrixanalízis

Last updated: 2021. március 12.

 Budapest University of Technology and Economics Faculty of Electrical Engineering and Informatics
 Course ID Semester Assessment Credit Tantárgyfélév VIMAD569 tavasz 4/0/0/v 5 1/1
3. Course coordinator and department Dr. Pach Péter Pál,
Web page of the course http://cs.bme.hu/~ppp/ma/index-en.html
4. Instructors
 Dr. Friedl Katalin associate professor Department of Computer Science and Information Theory Dr. Pach Péter Pál associate professor Department of Computer Science and Information Theory
5. Required knowledge Linear algebra, mathematical analysis.
6. Pre-requisites
Ajánlott:
None.
7. Objectives, learning outcomes and obtained knowledge The students get a deeper insight into the theory of linear algebra. There will be a special emphasis on matrix functions, Jordan normal form, and their application for solving systems of differential equations. This way we would like to deepen the student’s knowledge and understanding according to the demands of other subjects.
8. Synopsis

1.    Special matrix products; Rank One Decomposition, linear combination of rank-one matrices

2.    Matrix inversion

3.    Rank of a matrix, minimal rank-one decomposition

4.    Rank theorems, equivalent transformations

5.    Sylvester’s law of nullity

6.    Special matrices and their inverse

7.    Inverse of a modified matrix

9.    Generalized inverse

10.  Theory of systems of linear equations and their solution

11.  Systems of linear equations with a quadratic coefficient matrix

12.  Linear transformations; bilinear and quadratic forms

13.  The eigenvalue problem of matrices

14.  Diagonalizable transformations; Hermitian and unitary transformations

15.  Spectral decomposition of matrices; unitarily diagonalizable matrices

16.  Cayley - Hamilton theorem and its refinement

17.  Matrix functions and reduction to matrix polynomials

18.  (Matrix-valued) Lagrange-polynomials and their properties

19.  Determining the Lagrange matrix polynomials with the help of the adjugate of the characteristic matrix

20.  Calculating matrix functions with Hermite matrix polynomials; non-diagonalizable matrices

21.  Properties of the Hermite matrix polynomials

22.  Transforming nilpotent matrices into Jordan normal form

23.  Canonical form of functions of nilpotent matrices

24.  Canonical form of matrix functions

25.  Theory of elementary divisors

26.  Application of matrix functions in the theory of systems of linear differential equations

27.   Solving linear systems of differential equations

9. Method of instruction 4 hours of lecture per week.
10. Assessment

Signature: 1 homework.

Final: oral exam.

12. Consultations In office hours or by appointment.
13. References, textbooks and resources

Rózsa Pál: Lineáris algebra és alkalmazásai. 3. átdolgozott kiadás. Tankönyvkiadó, Budapest, 1991.

H. Golub – C.F. Van Loan: Matrix Computations, The John Hopkins University Press, 1989.

14. Required learning hours and assignment
 Kontakt óra 56 Félévközi készülés órákra 28 Felkészülés zárthelyire 0 Házi feladat elkészítése 16 Kijelölt írásos tananyag elsajátítása 10 Vizsgafelkészülés 40 Összesen 150
15. Syllabus prepared by Dr. Rózsa Pál, professor, Department of Computer Science and Information Theory