The spectral norm of a Horadam circulant matrix
Jorma K. Merikoski, Pentti Haukkanen, Mika Mattila, Timo Tossavainen
aa r X i v : . [ m a t h . N T ] M a y The spectral norm of a Horadam circulant matrix
Jorma K. Merikoski a , Pentti Haukkanen a , Mika Mattila b , Timo Tossavainen c, ∗ a Faculty of Natural Sciences, FI-33014 University of Tampere, Finland b Department of Mathematics, Tampere University of Technology, P.O. Box 553, FI-33101Tampere, Finland c Department of Arts, Communication and Education, Lulea University of Technology,SE-97187 Lulea, Sweden
Abstract
Let a , b , p , q be integers and ( h n ) defined by h = a , h = b , h n = ph n − + qh n − , n = 2 , , . . . . Complementing to certain previously known results, we study thespectral norm of the circulant matrix corresponding to h , . . . , h n − . Keywords:
Circulant matrix, Fibonacci sequence, Horadam sequence, Lucassequence, Spectral norm
1. Introduction
Throughout this paper, let a, b, p, q ∈ Z . We define the Horadam sequence ( h n ) = ( h n ( a, b ; p, q )) via h = a, h = b,h n = ph n − + qh n − , n = 2 , , . . . . We also use the following abbreviations:( f n ) = ( h n (0 ,
1; 1 , f n ) = ( h n (0 , p, q )), a generalization of the Fibonacci sequence;( l n ) = ( h n (2 ,
1; 1 , ∗ Corresponding author
Email addresses: [email protected] (Jorma K. Merikoski), [email protected] (Pentti Haukkanen), [email protected] (Mika Mattila), [email protected] (Timo Tossavainen)
Preprint submitted to Journal of L A TEX Templates October 3, 2018 ˜ l n ) = ( h n (2 , p ; p, q )), a generalization of the Lucas sequence.Some references call (˜ l n ) the Lucas sequence. In order to keep the languagesimple, we follow the custom in [7, p. 8] and call the sequence of Lucas numbersbriefly the Lucas sequence.For n ≥
1, we write f = ( f , . . . , f n − ) , ˜ f = ( ˜ f , . . . , ˜ f n − ) , l = ( l , . . . , l n − ) , ˜ l = (˜ l , . . . , ˜ l n − ) , h = ( h , . . . , h n − ) . Let x = ( x , . . . x n − ) ∈ R n . The corresponding circulant matrix C ( x ) isdefined as C ( x ) = x x . . . x n − x n − x n − x . . . x n − x n − x n − x n − . . . x n − x n − ... ... ... ... ... x x . . . x x x x . . . x n − x . We let k · k stand for the spectral norm. Our problem is to compute k C ( h ) k under suitable assumptions. Recently, Kocer et al. [6], ˙Ipek [5], Liu [8], andBah¸si [1] have already studied this question. We will survey their results inSection 2 and give further results in Sections 3 and 4. Finally, we will completeour paper with some remarks in Section 5.
2. Previous results
Let us first study the eigenvalues and singular values of C ( x ). Theorem 1.
The eigenvalues of C ( x ) are λ i = n − X j =0 x j ω − ij , i = 1 , . . . , n, where ω is the n ’th primitive root of unity. roof. See [2, Theorem 3.2.2].
Corollary 1.
The singular values of C ( x ) are σ i = (cid:12)(cid:12)(cid:12) n − X j =0 x j ω − ij (cid:12)(cid:12)(cid:12) , i = 1 , . . . , n. Therefore k C ( x ) k = max ≤ i ≤ n (cid:12)(cid:12)(cid:12) n − X j =0 x j ω − ij (cid:12)(cid:12)(cid:12) . Proof.
Since C ( x ) is normal, its singular values are the absolute values of eigen-values.Applying this corollary, Kocer et al. [6, Theorem 2.2] proved that k C ( h ) k = max ≤ i ≤ n − (cid:12)(cid:12)(cid:12) h n + ( pa − b + qh n − ) ω − i − aqω − i + pω − i − (cid:12)(cid:12)(cid:12) . The maximization problem restricts the use of this formula. The same authorsalso proved [6, Corollary 2.3] that k C ( h ) k = h n + qh n − + ( p − a − p + q − , (1)assuming that p, q ≥ b = 1. Doing so, they suppose nothing on a , butapparently a ≥ n = 1.)Further, ˙Ipek [5, Theorem 1] proved (independently of (1)) that k C ( f ) k = f n +1 − k C ( l ) k = f n +2 + f n − . Liu [8, Theorem 9] extended (1) to k C ( h ) k = h n + qh n − + ( p − a − bp + q − , (2)whenever p + q = 1, and to k C ( h ) k = qh n − + ( n − qa + b ) + aq + 1 , p + q = 1, but assumed nothing about a, b, p, q .Bah¸si [1, Theorem 2.1] proved (independently of (2)) that, if p, q ≥
1, then k C (˜ f ) k = ˜ f n + q ˜ f n − − p + q − k C (˜ l ) k = ˜ l n + q ˜ l n − + p − p + q − .
3. Computation of k C(h) k , h ≥ We first take a more general viewpoint and verify a theorem that applies alsoto other matrices than circulant ones or those having elements from a recurrencesequence. If a matrix A and a vector x are entrywise nonnegative (respectively,positive), we denote A ≥ O and x ≥ (respectively, A > O and x > ). Welet λ ( A ) denote the Perron root of a square matrix A ≥ O . Theorem 2.
Assume that an n × n matrix A ≥ O has all row sums and columnsums equal; let s be their common value. Then λ ( A ) = k A k = s .Proof. Denoting e = (1 , . . . , ∈ R n , we have Ae = A T e = s e . So, s is aneigenvalue of A and A T , and e is a corresponding eigenvector. Since e > ,actually s = λ ( A ) = λ ( A T ), see [4, Theorem 8.3.4]. Because A T Ae = A T s e = s A T e = s e , we similarly see that s = λ ( A T A ) = k A k . Corollary 2. If x = ( x , . . . , x n − ) ≥ , then k C ( x ) k = x + · · · + x n − . In order to apply this corollary in the case x = h , we must compute h + · · · + h n − . 4 emma 1. If p + q = 1 , then h + · · · + h n − = h n + qh n − + ( p − a − bp + q − . (3) If p + q = 1 and p = 2 , then h + · · · + h n − = qh n − + ( n − qa + b ) + aq + 1 . (4) If p = 2 and q = − , then h + · · · + h n − = n h n − + a . (5) Proof.
Claim (3) is equivalent to [3, Equation (3.5)] and to [8, Lemma 5(1)].Claim (4) is equivalent to [8, Lemma 5(2)]. Claim (5) is trivial, because thesequence ( h n ) is arithmetic.We have now proved the following theorem. Theorem 3. If h ≥ , then k C ( h ) k = h + · · · + h n − , where h + · · · + h n − is as in Lemma .
4. Generalization of Theorem 3
Can the assumption h ≥ be weakened? Again, we begin by taking a moregeneral viewpoint. For m ∈ Z , we set m n = m − j mn k n. Theorem 4.
Let x = ( x , . . . , x n − ) ∈ R n . If n − X i =0 x i x ( i + j − n ≥ for all j = 1 , . . . , n , then k C ( x ) k = | x + · · · + x n − | . (6)5 roof. Write B = ( b ij ) = C ( x ) T C ( x ). Letting c , . . . , c n to denote the columnvectors of C ( x ), we have b j = c · c j = n − X i =0 x i x ( i + j − n for all j = 1 , . . . , n . So, the first row of B is nonnegative. Summing its elementsgives us r = n X j =1 n − X i =0 x i x ( i + j − n = n − X i =0 x i n X j =1 x ( i + j − n = (cid:16) n − X i =0 x i (cid:17) . The last equation follows from the fact that (cid:8) i n , . . . , ( i + n − n (cid:9) = { , . . . , n − } for all i = 0 , . . . , n − B . Con-sequently, B ≥ O with row sums r = · · · = r n = (cid:16) n − X i =0 x i (cid:17) . Since B is symmetric, every of its column sums has this value, too. ApplyingTheorem 2 to B , we therefore obtain k C ( x ) k = λ ( B ) = (cid:16) n − X i =0 x i (cid:17) , and (6) follows. Corollary 3. If n − X i =0 h i h ( i + j − n ≥ for all j = 1 , . . . , n , then k C ( h ) k = | h + · · · + h n − | , where h + · · · + h n − is as in Lemma . . Concluding remarks In Section 2, we saw that, in the previous literature, k C ( h ) k is computedunder various assumptions on h . For example, in [6], the Horadam numbers wereinvolved requiring that a ≥ b = 1 and p, q ≥
1. We assumed first only that h ≥ , and then, even more generally, that (7) holds. As byproducts, Corollary2 and Theorem 4 provided us with the corresponding results on k C ( x ) k , too.We also mention that Yazlik and Taskara [9] defined the notion of a gener-alized k -Horadam sequence ( H k,n ) n ∈ N . In fact, Liu [8] ended up with (2) bystudying a circulant matrix corresponding to such a sequence. However, since k is fixed in [9, Definition 1], this sequence is nothing but an ordinary Horadamsequence ( h n ) = ( h n ( a, b ; p, q )) with p = f ( k ) and q = g ( k ). ReferencesReferences [1] M. Bah¸si, On the norms of circulant matrices with the generalized Fibonacciand Lucas numbers, TWMS Journal of Pure and Applied Mathematics 6(2015) 84–92.[2] P. J. Davis, Circulant Matrices, Wiley, 1979.[3] A. F. Horadam, Basic properties of a certain generalized sequence of num-bers, The Fibonacci Quarterly 3 (1965) 161–176.[4] R. A. Horn, C. R. Johnson, Matrix Analysis, Second Edition, CambridgeUniv. Pr., 2013.[5] A. ˙Ipek, On the spectral norms of circulant matrices with classical Fibonacciand Lucas numbers entries, Applied Mathematics and Computation 217(2011) 6011–6012.[6] E. G. Kocer, T. Mansour, N. Tuglu, Norms of circulant and semicirculantmatrices with Horadam numbers, Ars Combinatoria 85 (2007) 353–359.77] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley, 2001.[8] L. Liu, On the spectrum and spectral norms of r -circulant matrices with gen-eralized k -Horadam number entries, International Journal of ComputationalMathematics 2014, Art. ID 795175, 6 pp.[9] Y. Yazlik, N. Taskara, A note on generalized kk