The article discusses the matrices of the form A(n)(1), A(n)(m), A(N)(m), whose inverses are: tridiagonal matrix A(n)(-1) (n - dimension of the A(N)(-m) matrix), banded matrix A(n)(-m) (m is the half-width band of the matrix) or block-tridiagonal matrix A(N)(-m) (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A(n)(1), A(n)(m), A(N)(m) have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given.
Publication Name (dc.title) | Matrices Whose Inverses are Tridiagonal, Band or Block-Tridiagonal and Their Relationship with the Covariance Matrices of a Random Markov Process |
Author/s (dc.contributor.yazarlar) | Ulan Brimkulov |
Publication type (dc.type) | Konferans Bildirisi |
Language (dc.language) | İngilizce |
Publication year (dc.date.issued) | 2019 |
National/International (dc.identifier.ulusaluluslararasi) | Uluslararası |
Source (dc.relation.journal) | Filomat |
Additional source name / Conference information (dc.identifier.kaynakadiekbilgi) | 8th International Conference on Mathematical Analysis, Differential Equation and Applications (MADEA).- Cholpon Ata, KYRGYZSTAN.- JUN 17-23, 2018 |
Number (dc.identifier.issue) | 5 |
Volume/Issue (dc.identifier.volume) | 33 |
Page (dc.identifier.startpage) | 1335-1352 |
ISSN/ISBN (dc.identifier.issn) | ISSN: 0354-5180; Online ISSN: 2406-0933 |
Publisher (dc.publisher) | Prirodno-matematički fakultet, University of Niš, Serbia |
Databases (dc.contributor.veritaban) | Web of Science Core Collection |
Databases (dc.contributor.veritaban) | Kaynak web sitesi |
Databases (dc.contributor.veritaban) | Scopus |
Index Type (dc.identifier.index) | CPCI-S |
Index Type (dc.identifier.index) | Scopus |
Impact Factor (dc.identifier.etkifaktoru) | 0,789 / 2018-WOS / 5 Year: 0,852 |
Abstract (dc.description.abstract) | The article discusses the matrices of the form A(n)(1), A(n)(m), A(N)(m), whose inverses are: tridiagonal matrix A(n)(-1) (n - dimension of the A(N)(-m) matrix), banded matrix A(n)(-m) (m is the half-width band of the matrix) or block-tridiagonal matrix A(N)(-m) (N = n x m - full dimension of the block matrix; m - the dimension of the blocks) and their relationships with the covariance matrices of measurements with ordinary (simple) Markov Random Processes (MRP), multiconnected MRP and vector MRP, respectively. Such covariance matrices frequently occur in the problems of optimal filtering, extrapolation and interpolation of MRP and Markov Random Fields (MRF). It is shown, that the structures of the matrices A(n)(1), A(n)(m), A(N)(m) have the same form, but the matrix elements in the first case are scalar quantities; in the second case matrix elements represent a product of vectors of dimension m; and in the third case, the off-diagonal elements are the product of matrices and vectors of dimension m. The properties of such matrices were investigated and a simple formulas of their inverses were found. Also computational efficiency in the storage and the inverse of such matrices have been considered. To illustrate the acquired results, an example on the covariance matrix inversions of two-dimensional MRP is given. |
URL (dc.rights) | http://www.pmf.ni.ac.rs/filomat-content/2019/33-5/33-5-7-8852.pdf |
DOI (dc.identifier.doi) | 10.2298/FIL1905335B |
Faculty / Institute (dc.identifier.fakulte) | Mühendislik Fakültesi |
Department (dc.identifier.bolum) | Bilgisayar Mühendisliği Bölümü |
Author(s) in the Institution (dc.contributor.author) | Ulan BRİMKULOV |
Kayıt No (dc.identifier.kayitno) | BL2A3D4DAC |
Record Add Date (dc.date.available) | 2019-11-29 |
Notes (Publication year) (dc.identifier.notyayinyili) | 2019 |
Wos No (dc.identifier.wos) | WOS:000496192400008 |
Subject Headings (dc.subject) | best linear unbiased estimates (BLUE) |
Subject Headings (dc.subject) | simple (ordinary connected) Markov process |
Subject Headings (dc.subject) | multiply connected (m-connected) Markov process |
Subject Headings (dc.subject) | vector (m-dimensional) Markov process |
Subject Headings (dc.subject) | random field filtering and parametric identification |
Subject Headings (dc.subject) | tridiagonal matrices |
Subject Headings (dc.subject) | banded matrices |
Subject Headings (dc.subject) | block-tridiagonal matrices |