Araştırma Makalesi
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Factorizations of Some Variants of a Statistical Matrix

Yıl 2024, Cilt: 16 Sayı: 1, 229 - 239, 30.06.2024
https://doi.org/10.47000/tjmcs.1428063

Öz

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.

Kaynakça

  • 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.
Yıl 2024, Cilt: 16 Sayı: 1, 229 - 239, 30.06.2024
https://doi.org/10.47000/tjmcs.1428063

Öz

Kaynakça

  • 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.
Toplam 18 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Cebir ve Sayı Teorisi
Bölüm Makaleler
Yazarlar

Gonca Kızılaslan 0000-0003-1816-6095

Harun Şahin 0000-0003-1007-7223

Yayımlanma Tarihi 30 Haziran 2024
Gönderilme Tarihi 29 Ocak 2024
Kabul Tarihi 10 Haziran 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 16 Sayı: 1

Kaynak Göster

APA Kızılaslan, G., & Şahin, H. (2024). Factorizations of Some Variants of a Statistical Matrix. Turkish Journal of Mathematics and Computer Science, 16(1), 229-239. https://doi.org/10.47000/tjmcs.1428063
AMA Kızılaslan G, Şahin H. Factorizations of Some Variants of a Statistical Matrix. TJMCS. Haziran 2024;16(1):229-239. doi:10.47000/tjmcs.1428063
Chicago Kızılaslan, Gonca, ve Harun Şahin. “Factorizations of Some Variants of a Statistical Matrix”. Turkish Journal of Mathematics and Computer Science 16, sy. 1 (Haziran 2024): 229-39. https://doi.org/10.47000/tjmcs.1428063.
EndNote Kızılaslan G, Şahin H (01 Haziran 2024) Factorizations of Some Variants of a Statistical Matrix. Turkish Journal of Mathematics and Computer Science 16 1 229–239.
IEEE G. Kızılaslan ve H. Şahin, “Factorizations of Some Variants of a Statistical Matrix”, TJMCS, c. 16, sy. 1, ss. 229–239, 2024, doi: 10.47000/tjmcs.1428063.
ISNAD Kızılaslan, Gonca - Şahin, Harun. “Factorizations of Some Variants of a Statistical Matrix”. Turkish Journal of Mathematics and Computer Science 16/1 (Haziran 2024), 229-239. https://doi.org/10.47000/tjmcs.1428063.
JAMA Kızılaslan G, Şahin H. Factorizations of Some Variants of a Statistical Matrix. TJMCS. 2024;16:229–239.
MLA Kızılaslan, Gonca ve Harun Şahin. “Factorizations of Some Variants of a Statistical Matrix”. Turkish Journal of Mathematics and Computer Science, c. 16, sy. 1, 2024, ss. 229-3, doi:10.47000/tjmcs.1428063.
Vancouver Kızılaslan G, Şahin H. Factorizations of Some Variants of a Statistical Matrix. TJMCS. 2024;16(1):229-3.