On some properties of a generalized min matrix

Emrah Polatlı; Emrah Polatlı

doi:10.3934/math.20231336

AIMS Mathematics

2023, Volume 8, Issue 11: 26199-26212. doi: 10.3934/math.20231336

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Research article

On some properties of a generalized min matrix

Emrah Polatlı ^,

Department of Mathematics, Zonguldak Bülent Ecevit University, 67100, TURKEY

Received: 04 July 2023 Revised: 31 August 2023 Accepted: 07 September 2023 Published: 13 September 2023
MSC : 15A09, 15A23, 15A60, 15B05, 15B99

In this paper, we investigate a min matrix and obtain its $ LU $-decomposition, determinant, permanent, inverse, and norm properties. In addition, we obtain a recurrence relation provided by the characteristic polynomial of this matrix. Finally, we present an example to illustrate the results obtained.
- Frank matrix,
- min matrix,
- determinant,
- inverse,
- permanent,
- norm
Citation: Emrah Polatlı. On some properties of a generalized min matrix[J]. AIMS Mathematics, 2023, 8(11): 26199-26212. doi: 10.3934/math.20231336

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Abstract

In this paper, we investigate a min matrix and obtain its $ LU $-decomposition, determinant, permanent, inverse, and norm properties. In addition, we obtain a recurrence relation provided by the characteristic polynomial of this matrix. Finally, we present an example to illustrate the results obtained.

References

[1]	W. Frank, Computing eigenvalues of complex matrices by determinant evaluation and by methods of Danilewski and Wielandt, J. Soc. Indust. Appl. Math., 6 (1958), 378–392. http://dx.doi.org/10.1137/0106026 doi: 10.1137/0106026
[2]	J. Hake, A remark on Frank matrices, Computing, 35 (1985), 375–379. http://dx.doi.org/10.1007/BF02240202 doi: 10.1007/BF02240202
[3]	J. Varah, A generalization of the Frank matrix, SIAM Journal on Scientific and Statistical Computing, 7 (1986), 835–839.
[4]	P. Eberlein, A note on the matrices denoted $B_{n}^{\ast }$, SIAM J. Appl. Math., 20 (1971), 87–92. http://dx.doi.org/10.1137/0120012 doi: 10.1137/0120012
[5]	E. Kılıç, T. Arıkan, Studying new generalizations of max-min matrices with a novel approach, Turk. J. Math., 43 (2019), 2010–2024. http://dx.doi.org/10.3906/mat-1811-95 doi: 10.3906/mat-1811-95
[6]	C. Kızılateş, N. Terzioğlu, On $r$-min and $r$-max matrices, J. Appl. Math. Comput., 68 (2022), 4559–4588. http://dx.doi.org/10.1007/s12190-022-01717-y doi: 10.1007/s12190-022-01717-y
[7]	Y. Liu, Z. Jiang, X. Jiang, Two types of interesting Fibonacci-min matrices, Adv. Appl. Discret. Math., 24 (2020), 13–25. http://dx.doi.org/10.17654/DM024010013 doi: 10.17654/DM024010013
[8]	S. Wang, Z. Jiang, Y. Zheng, Determinants, inverses and eigenvalues of two symmetric positive definite matrices with Pell and Pell-Lucas numbers, Adv. Differ. Equ. Contr., 22 (2020), 83–95. http://dx.doi.org/10.17654/DE022020083 doi: 10.17654/DE022020083
[9]	Q. Meng, X. Jiang, Z. Jiang, Interesting determinants and inverses of skew Loeplitz and Foeplitz matrices, J. Appl. Anal. Comput., 11 (2021), 2947–2958. http://dx.doi.org/10.11948/20210070 doi: 10.11948/20210070
[10]	Q. Meng, Y. Zheng, Z. Jiang, Exact determinants and inverses of (2, 3, 3)-Loeplitz and (2, 3, 3)-Foeplitz matrices, Comp. Appl. Math., 41 (2022), 35. http://dx.doi.org/10.1007/s40314-021-01738-6 doi: 10.1007/s40314-021-01738-6
[11]	Q. Meng, Y. Zheng, Z. Jiang, Determinants and inverses of weighted Loeplitz and weighted Foeplitz matrices and their applications in data encryption, J. Appl. Math. Comput., 68 (2022), 3999–4015. http://dx.doi.org/10.1007/s12190-022-01700-7 doi: 10.1007/s12190-022-01700-7
[12]	E. Mersin, M. Bahşi, A. Maden, Some properties of generalized Frank matrices, Mathematical Sciences and Applications E-Notes, 8 (2020), 170–177. http://dx.doi.org/10.36753/mathenot.672621 doi: 10.36753/mathenot.672621
[13]	R. Mathias, The spectral norm of a nonnegative matrix, Linear Algebra Appl., 139 (1990), 269–284. http://dx.doi.org/10.1016/0024-3795(90)90403-Y doi: 10.1016/0024-3795(90)90403-Y
[14]	M. Bahşi, On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers, TWMS J. Pure Appl. Math., 6 (2015), 84–92.
[15]	M. Bahşi, On the norms of $r$-circulant matrices with the hyperharmonic numbers, J. Math. Inequal., 10 (2016), 445–458. http://dx.doi.org/10.7153/jmi-10-35 doi: 10.7153/jmi-10-35
[16]	Z. Jiang, Z. Zhou, A note on spectral norms of even-order $r$-circulant matrices, Appl. Math. Comput., 250 (2015), 368–371. http://dx.doi.org/10.1016/j.amc.2014.11.020 doi: 10.1016/j.amc.2014.11.020
[17]	C. Kızılateş, N. Tuğlu, On the bounds for the spectral norms of geometric circulant matrices, J. Inequal. Appl., 2016 (2016), 312. http://dx.doi.org/10.1186/s13660-016-1255-1 doi: 10.1186/s13660-016-1255-1
[18]	S. Shen, J. Cen, On the bounds for the norms of $r$-circulant matrices with Fibonacci and Lucas numbers, Appl. Math. Comput., 216 (2010), 2891–2897. http://dx.doi.org/10.1016/j.amc.2010.03.140 doi: 10.1016/j.amc.2010.03.140
[19]	S. Shen, J. Cen, On the spectral norms of $r$-circulant matrices with the $k$-Fibonacci and $k$-Lucas numbers, Int. J. Contemp. Math. Sci., 5 (2010), 569–578.
[20]	B. Shi, C. Kızılateş, Some spectral norms of RFPRLRR circulant matrices, Filomat, 37 (2023), 4221–4238. http://dx.doi.org/10.2298/FIL2313221S doi: 10.2298/FIL2313221S
[21]	S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160 (2005), 125–132. http://dx.doi.org/10.1016/j.amc.2003.08.126 doi: 10.1016/j.amc.2003.08.126
[22]	E. Polatlı, On the bounds for the spectral norms of $r$-circulant matrices with a type of Catalan triangle numbers, J. Sci. Arts, 3 (2019), 575–586.
[23]	E. Polatlı, On geometric circulant matrices whose entries are bi-periodic Fibonacci and bi-periodic Lucas numbers, Universal Journal of Mathematics and Applications, 3 (2020), 102–108. http://dx.doi.org/10.32323/ujma.669276 doi: 10.32323/ujma.669276
[24]	B. Radičić, On $k$-circulant matrices involving the Pell-Lucas (and the modified Pell) numbers, Comput. Appl. Math., 40 (2021), 111. http://dx.doi.org/10.1007/s40314-021-01473-y doi: 10.1007/s40314-021-01473-y
[25]	B. Radičić, On geometric circulant matrices with geometric sequence, Linear Multilinear Algebra, in press. http://dx.doi.org/10.1080/03081087.2023.2188156
[26]	B. Radičić, The inverse and the Moore-Penrose inverse of a $k$-circulant matrix with binomial coefficients, Bull. Belg. Math. Soc. Simon Stevin, 27 (2020), 29–42. http://dx.doi.org/10.36045/bbms/1590199301 doi: 10.36045/bbms/1590199301
[27]	R. Brualdi, P. Gibson, Convex polyhedra of doubly stochastic matrices. Ⅰ. applications of the permanent function, J. Comb. Theory A, 22 (1977), 194–230. http://dx.doi.org/10.1016/0097-3165(77)90051-6 doi: 10.1016/0097-3165(77)90051-6
[28]	E. Mersin, M. Bahşi, Sturm theorem for the generalized Frank matrix, Hacet. J. Math. Stat., 50 (2021), 1002–1011. http://dx.doi.org/10.15672/hujms.773281 doi: 10.15672/hujms.773281
[29]	E. Mersin, Sturm's theorem for min matrices, AIMS Mathematics, 8 (2023), 17229–17245. http://dx.doi.org/10.3934/math.2023880 doi: 10.3934/math.2023880

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