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PřF:M6370 Special matrices - Course Information

## M6370 Special matrices

**Faculty of Science**

Spring 2005

**Extent and Intensity**- 2/1/0. 3 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
**Teacher(s)**- prof. RNDr. Ladislav Skula, DrSc. (lecturer)

Mgr. Jiří Zelinka, Dr. (seminar tutor) **Guaranteed by**- prof. RNDr. Ivanka Horová, CSc.

Department of Mathematics and Statistics - Departments - Faculty of Science

Contact Person: prof. RNDr. Ladislav Skula, DrSc. **Timetable**- Wed 13:00–13:50 N21, Fri 8:00–9:50 N41
**Prerequisites**-
**M2110**Linear Algebra II

Basic knowledge of matrix analysis. Elements of mathematical statistics. **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Mathematics - Economics (programme PřF, M-AM)

**Course objectives**- The basic notions of matrix theory: block matrices, permutation matrices, Kronecker product. Differential operators for matrices. The Fourier analysis: the Fourier matrix,the discrete Fourier transform, sampling of peridic function (discretisation). The discrete convolution. Fast algorithms of the discrete Fourier transform and the convolution. General inverse matrices: generalized inverse, the Moore-Penrose inverse (existence and methods of calculations), solution to the system of linear equations by means of generalized inverses, least square method. Application of linear algebra in mathematical economy, Leontieff model. Application of matrix analysis in statistics: covariance and correlation matrices, Wishart matrix, linear model.
**Syllabus**- Introduction of new notions requested in the field of special matrices: block matrices and operations with these matrices, direct sums, Kronecker product of matrices, permutation matrices, forward shift permutation, the eigenvalues of permutation matrices. Differntial operators for matrices. Fourier analysis: the Fourier matrix, the eigenvalues of Fourier matrix with multiplicities, the discrete Fourier transform and its inverse, Fourier matrices as Kronecker products, Fast Fourier Transform, the Walsh-Hadamard transform, sampling of peridic functions.Discrete convolutions. Fast methods of calculations of the discrete Fourier transform and the discrete convolutions. The Fourier coefficients of a vector. The inner product space of continuous functions, Fourier expansion. General inverses of matrices: right and left inverses, their determination and existence, generalized inverses, their description and calculation, minor algorithm, application of generalized inverses to a system of linear equation, linearly independent solutions. Some theorems on the rank of a matrix. The Moore-Penrose inverse, existence and uniqueness, the methods of calculations (rank factorization theorem, Greville algorithm). The singular value decomposition theorem. The least square solution, minimum norm. Application of linear algebra in mathematical economy: Leontieff input-output model, non-negative matrices and their use in Leontieff model. Application in mathematical statistics: variance-covariance matrix and itslinear transform, correlation matrix, singular covariance matrices and quadratic forms, Wishart matrix. Linear model, unbiased estimator, use of generalized inverses (best linear estimator).

**Literature**- [1] J.Anděl, Matematická statistika,SNTL, Praha,1985
- P.J.Davis, Circulant Matrices, New York,1994
- P.Horák, Algebra a teoretická aritmetika I., Brno,1991, MU
- S.S.Searle, Matrix Algebra Useful for Statistics, New York,1982
- F.Šik, Lineární algebra zaměřená na numerickou analýzu, Brno 1998, MU

**Assessment methods**(in Czech)- Výuka: přednáška, cvičení. Zkouška: písemná a ústní.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- The course is taught annually.

- Enrolment Statistics (Spring 2005, recent)
- Permalink: https://is.muni.cz/course/sci/spring2005/M6370