Factorizations of Some Variants of a Statistical Matrix
Year 2024,
Volume: 16 Issue: 1, 229 - 239, 30.06.2024
Gonca Kızılaslan
Harun Şahin
Abstract
In this article, we define eight orthogonal matrices which are strongly related with the well known Helmert matrix. We obtain $LU$ factorizations by giving explicit closed-form formulas of the entries of $L$ and $U$. We also factor matrices by expressing them in terms of diagonal matrices.
References
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Year 2024,
Volume: 16 Issue: 1, 229 - 239, 30.06.2024
Gonca Kızılaslan
Harun Şahin
References
- Akbıyık, M., Yamaç¸ Akbıyık, S., Yılmaz, F., On linear algebra of one type of symmetric matrices with harmonic Fibonacci entries, Notes on Number Theory and Discrete Mathematics, 28 (3)(2022), 399–410.
- Andelic, M., da Fonseca, C.M., Yılmaz, F., The bi-periodic Horadam sequence and some perturbed tridiagonal 2−Toeplitz matrices: A unified approach, Heliyon, 8(2)(2022).
- Akkus, I., Kizilaslan, G., Generalization of a statistical matrix and its factorization, Communications in Statistics-Theory and Methods, 50(4)(2021), 963–978.
- Birregah, B., Doh, P.K., Adjallah, K.H., A systematic approach to matrix forms of the Pascal triangle: The twelve triangular matrix forms and relations, European Journal of Combinatorics, 31(5)(2010), 1205–1216.
- Clarke, B.R., Linear Models: The Theory and Application of Analysis of Variance, Wiley, 2008.
- Doh, P.K., Adjallah, K.H., Birregah, B., Thirty-six full matrix forms of the Pascal triangle: Derivation and symmetry relations, Scientific African, 13(2021), e00932.
- Farhadian, R., A note on a generalization of a statistical matrix, Communications in Statistics–Theory and Methods, 50(12)(2021), 2938–2946.
- Fonseca, C., Kizilates, C., Terzioglu, N., A second-order difference equation with sign-alternating coefficients, Kuwait Journal of Science, 50(2A)(2023), 1–8.
- Fonseca, C., Kizilates, C., Terzioğlu N., A new generalization of min and max matrices and their reciprocals counterparts, Filomat, 38(2)(2024), 421–435.
- Gentie, J.E., Numerical Linear Algebra for Application in Statistics, Springer, 1998.
- Helmert, F.R., Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit, Astronom. Nachr., 88(1876), 115–132.
- Hürlimann, W., Generalized Helmert-Ledermann orthogonal matrices and rom simulation, Linear Algebra Appl., 439(7) (2013), 1716–1729.
- Irwin, J.O., On the distribution of a weighted estimate of variance and on analysis of variance in certain cases of unequal weighting, J. Roy. Statist. Soc. Ser., 105(1942), 115–118.
- Kızılateş, C., Terzioğlu, N., On r−min and r−max matrices, Journal of Applied Mathematics and Computing, 68(6)(2022), 4559–4588.
- Lancaster, H.O., The Helmert matrices, Amer. Math. Monthly, 72(1965), 4–12.
- Pearson, K., On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling, Philos. Mag., 50(5)(1900), 157–175.
- Seber, G.A.F., A Matrix Handbook for Statisticians, Wiley, 2007.
- Shi, B., Kızılates¸, C., A new generalization of the Frank matrix and its some properties, Computational and Applied Mathematics, 43(1)(2024), 19.