# The upper bound estimation for the spectral norm of $r$-circulant and symmetric $r$-circulant matrices with the Padovan sequence

Volume 9, Issue 1, pp 92--101
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### Authors

Wutiphol Sintunavarat - Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathumthani 12121, Thailand.

### Abstract

In this paper, we gives an upper bound estimation of the spectral norm for matrices $A$ and $B$ such that the entries in the first row of $n\times n$ $r$-circulant matrix $A = Circ_r(a_1; a_2; ...; a_n)$ and $n\times n$ symmetric $r$-circulant matrix $B = SCirc_r(a_1; a_2; ...; a_n)$ are $a_i = P_i$ or $a_i = P^2 _i$ or $a_i = P_{i-1}$ or $a_i = P^2_{i-1}$, where $\{P_i\}^\infty_{i =0}$ is Padovan sequence. At the last section, some illustrative numerical example is furnished which demonstrate the validity of the hypotheses and degree of utility of our results.

### Share and Cite

##### ISRP Style

Wutiphol Sintunavarat, The upper bound estimation for the spectral norm of $r$-circulant and symmetric $r$-circulant matrices with the Padovan sequence, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 1, 92--101

##### AMA Style

Sintunavarat Wutiphol, The upper bound estimation for the spectral norm of $r$-circulant and symmetric $r$-circulant matrices with the Padovan sequence. J. Nonlinear Sci. Appl. (2016); 9(1):92--101

##### Chicago/Turabian Style

Sintunavarat, Wutiphol. "The upper bound estimation for the spectral norm of $r$-circulant and symmetric $r$-circulant matrices with the Padovan sequence." Journal of Nonlinear Sciences and Applications, 9, no. 1 (2016): 92--101

### Keywords

• r-circulant matrices
• symmetric r-circulant matrices

•  15A15
•  15A18
•  15A24
•  15A48

### References

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• [2] R. Mathias, The spectral norm of a nonnegative matrix, Linear Algebra Appl., 139 (1990), 269-284.

• [3] S. Shen, J. Cen , On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 216 (2010), 2891-2897.

• [4] S. Solak , On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160 (2005), 125-132.

• [5] S. Solak, Erratum to ''On the norms of circulant matrices with the Fibonacci and Lucas numbers'', [Appl. Math. Comput. 160 (2005) 125-132], Appl. Math. Comput., 190 (2007), 1855-1856.