Triangular Truncation of k-Fibonacci and k-Lucas Circulant Matrices
Triangular Truncation of k-Fibonacci and k-Lucas Circulant Matrices John Dixon, Michael Goldenberg, Ben Mathes and Justin Sukiennik September 11, 2013
Abstract We prove a general theorem that gives tight bounds on the spectral norms of triangularly truncated k-Fibonacci and k-Lucas circulant matrices. The bounds are good enough to enable the calculation of the limit ||C|| , ||τ (C)|| as the dimension n approaches infinity, where τ (C) denotes the triangular truncation of C, and C is any n × n circulant matrix built using a sequence (si ) satisfying si = ksi−1 + si−2 . In particular, we have that this limit is equal to the golden ratio, if C is built using either the ordinary Fibonacci or Lucas sequence.
1
Introduction
The k-Fibonacci and k-Lucas sequences are second order recursive sequences satisfing fi−2 + kfi−1 = fi , for all integers i ≥ 2, the two sequences determined by their initial values; the k-Fibonacci sequence begins with f0 = 0 and f1 = 1, while the k-Lucas sequence begins f0 = 2 and f1 = k. We will follow the notation used in [9] and denote the set of all such recursive sequences by R(k, 1). When k = 1 we obtain the ordinary Fibonacci and Lucas sequence. There has recently 0 0
AMS 2000 Classification numbers: Primary: 15B36 Secondary: 65F35 Key Words: Triangular truncation, Fibonacci numbers, spectral norm.
1
been a flurry of interest in the norms and singular values of various matrices built using the Fibonacci and Lucas sequences, and their generalizations defined above ([1], [2], [4], [5], [8], [14], [15], [16]). In [15], the focus is on spectral norm estimates for r-circulant matrices whose entries are generated via the k-Fibonacci or k-Lucas sequence for positive k. The r-circulant matrices are of the form a0 a1 . . . an−2 an−1 ran−1 a0 a1 ... an−2 ran−2 ran−1 a0 ... an−3 A= , .. .. .. . . . . . . ra1
ra2
. . . ran−1
a0
thus if r = 1, then A is the n × n circulant matrix determined by the sequence (ai ) (see [10] for a nice expository article on circulant matrices), while A becomes the triangular truncation of a circulant matrix when r = 0. Here is the main result of [15] in the case of the k-Fibonacci sequence: Theorem 1 [15] Let A be as above, with ai = fi , the k-Fibonacci sequence. 1. If |r| ≥ 1, then r
fn fn−1 |r| − |r|n (fn + |r|fn−1 ) ≤ ||A|| ≤ . k 1 − k|r| − |r|2
2. If |r| < 1, then r |r|
fn + fn−1 − 1 fn fn−1 ≤ ||A|| ≤ . k k
When r = 1, the matrix A is circulant and, since its entries are nonnegative, n−1 X fn−1 + fn − 1 ||A|| = fi = , k i=0
(see the preliminary section). While (2) of Theorem 1 gives no non-trivial lower bound for r = 0, it does say that triangular truncation of a circulant matrix whose entries are generated by a k-Fibonacci is contractive. The aim of this note is to give, for any non-zero k, tight estimates on the spectral norm of the triangularly truncated A.
2
2
Preliminaries
The space R(k, 1) is a two dimensional (real or) complex vector space with a natural basis consisting of two sequences of the form (1, τ, τ 2 , τ 3 , ...), corresponding to the two distinct roots λ and µ of the polynomial τ 2 −kτ −1. These two roots satisfy 0 < |µ| < 1 < |λ|, and we refer to λ as the k-golden ratio. As it happens, −1 < µ < 0 and 1 < λ for positive k, while λ < −1 and 0 < µ < 1 when k is negative. One has λ + µ = k and λµ = −1, and writing the Fibonacci and Lucas sequences, (fi ) and (li ), in terms of this basis gives the Binet formulae: fi =
λi − µi and li = λi + µi , λ−µ
for all integers i. Almost all of this can be found in [9]. An n × n circulant matrix corresponding to a sequence (ai ) is the matrix a0 a1 . . . an−2 an−1 an−1 a0 a1 . . . an−2 A = an−2 an−1 a0 . . . an−3 , .. .. . . .. .. . . a1 a2 . . . an−1 a0 and the set of all such circulant matrices forms a commutative algebra C of normal matrices, which is simultaneously diagonalizable. If 1 n−1 E = √ uij i,j=0 , n with u a primitive nth -root of unity, then E is the unitary that implements the Discrete Fourier Transform, and it is also the unitary that diagonalizes circulant matrices, i.e. E ∗ AE is a diagonal matrix with diagonal entries n−1 X
ai v i ,
i=0
3
one entry for each nth -root of unity v, e.g. v = ui for i = 0, . . . , n − 1. We will refer to E as the DFT unitary. Most of this paragraph can be found in [10]. 1 The singular values of a matrix A are the eigenvalues of (A∗ A) 2 , and traditionally they are listed in descending order s0 ≥ s1 ≥ . . . ≥ sm > 0, with m + 1 equal to the rank of A. The spectral norm ||A|| is defined to be qP m 2 s0 , and the Frobenius norm ||A||F is defined to be i=0 si . Writing the entries [aij ] of the matrix A, we get X ||A||2F = |aij |2 , ij
and ||A|| = sup ||Ax||, ||x||≤1
where ||x|| denotes the Euclidean norm of x. Thus, the spectral norm is an operator norm. When A is a diagonal matrix, then the singular values are the absolute values of the diagonal entries, and more generally, when A is normal, the singular values are the absolute values of the eigenvalues of A. In particular, if A is an n × n circulant matrix generated from a sequence (ai ), then the spectral norm of A is the supremum over |
n−1 X
ai v i |,
i=0
as v varies through the nth roots of unity. When (ai ) is non-negative, then this supremum is attained with v = 1, so that ||A|| =
n−1 X
ai .
i=0
In case F is the circulant corresponding to the k-Fibonacci sequence with positive k, we get n−1 X fn−1 + fn − 1 ||F || = fi = , k i=0
which is the upper bound of the corresponding r-circulant matrix obtained in [15], for |r| ≤ 1. It appears that the authors were unaware of this at the 4
time they wrote their paper, while for the ordinary Fibonacci sequence, this fact is explicitly noted in [8]. The circulant matrices are examples of Toeplitz operators: they are constant along each upper left to lower right diagonal. An infinite matrix [aij ] acting boundedly on `2 is an analytic Toeplitz operator if it is Toeplitz and triangular. As such, the matrix is given by a sequence (ai ), and, assuming it is upper triangular, is of the form a0 a1 a2 . . . 0 a0 a1 a2 . . . A= 0 0 ... ... ... . .. . ... P i The corresponding function f (z) = ∞ i=0 ai z is a bounded analytic function in the open unit disc, and with ||A|| denoting the operator norm of A, one has ||A|| = sup |f (z)|. |z|≤1
If A is an n × n truncated circulant matrix corresponding to (ai ), and it P i a is known that f (z) = ∞ i=0 i z is a bounded analytic function in the unit disc, then A is a compression of A, hence ||A|| ≤ ||A|| = sup |f (z)|. |z|≤1
These facts can be found in any text that treats Hardy spaces: see [13], [12], and [6]. A matrix space, endowed with the spectral norm, lets us describe the distance from a matrix A to a set K as inf ||A − K||.
k∈K
If (si ) denotes the sequence of singular values of A, arranged in decreasing order, then si is exactly the distance from A to the set of matrices of rank less than or equal to i (for i = 0, . . . , rank(A)). This fact can be found in [7]. We intend to use this fact in the following form: given A, we find a matrix O of rank one with ||A − O|| ≤ r. We conclude that si ≤ r for all i ≥ 1.
5
3
Results
The first thought one might have, presented with evidence that truncating particular circulant matrices is contractive, is that it might be a property of all circulant matrices rather than the particular ones. As it happens, this is not the case. In fact, the rate of growth of the truncation norm, when restricted to the set of circulant n × n matrices, is the same as the rate of growth on all n × n matrices (see [3] for a proof that the norm of the triangular trunctation operator, defined on the set of all n × n matrices, grows on the order of log n). Theorem 2 Triangular truncation is unbounded when restricted to circulant matrices. In fact, there exists n × n circulant matrices A with n ||τ (A)|| 1 ≥ log . ||A|| π 2 Proof. Let E denote the DFT unitary. Let {e1 , . . . , en } denote the standard basis of Cn . Our plan is to show that there exists an n×n diagonal matrix D0 of norm one such that n 1 | < E ∗ τ (ED0 E ∗ )Ee1 , e1 > | ≥ log . π 2 Observe that for a circulant C = circ(c0 , . . . , cn−1 ), the northwest entry of the matrix E ∗ τ (C)E is n−1
1X < E τ (C)Ee1 , e1 >= (n − i)ci . n ∗
i=0
Apply this to the circulant C, where ci = primitive nth root of unity, i.e. the circulant
1 n
Pn−1 j=0
aj uij , and with u a
C = EDE ∗ , with D = diag(a0 , . . . an−1 ), to get n−1 n−1 n−1 n−1 X 1 X 1 X X ij | < E τ (EDE )Ee1 , e1 > | = | 2 (n−i) aj u | = | 2 aj (n−i)uij |, n n ∗
∗
i=0
j=0
j=0
i=0
from which we see that the supremum over all diagonal contractions D is attained, and equal to 6
n−1 n−1 1 X X | < E τ (ED0 E )Ee1 , e1 > | = 2 | (n − i)uij |. n ∗
∗
j=0
i=0
Pn−1
Let’s look at a typical sum i=0 (n − i)uij with 1 ≤ j: let v 6= 1 denote any nth root of unity. We massage the sum to appear like the derivative of a finite geometric series, obtaining |
n−1 X
i
(n − i)v | = |(n + 1)
i=0
When v = e
n−1 X i=0
2πki n
i
v −
n−1 X
i
(i + 1)v | = |
i=0
n−1 X
(i + 1)v i | =
i=0
n . |1 − v|
, we have
2kπ , n and realizing the distances |1 − v| come in conjugate pairs, we deduce Pb n−1 c n(n+1) 1 n ∗ ∗ 2 | < E τ (ED0 E )Ee1 , e1 > | ≥ n2 + 2 k=1 2πki 2 |1−e n | n−1 Pb c n2 ≥ n12 n(n+1) + 2 k=12 2kπ 2 Pb n−1 c 1 n ≥ π1 k=12 k1 ≥ π log 2 |1 − v| ≤
We let Λ denote the n × n circulant matrix generated by the sequence and M the n × n circulant matrix generated by (µi ), (with λ and µ the roots of τ 2 − kτ − 1 as above, and λ the k- golden ratio). The general n × n circulant generated from a sequence in R(k, 1) is then (λi ),
C = αΛ + βM, with α, β ∈ C. Its triangular truncation we denote by T = τ (C) = ατ (Λ) + βτ (M ). Assume from now on that k is any non-zero real number. Lemma 3 Let (si ) denote the sequence of singular values of τ (Λ). Then si ≤
1 |λ| − 1
for all i ≥ 1. 7
Proof.
Let O denote the rank one n × n matrix 1 λ−1 −2 O= λ [1 λ λ2 . . . λn−1 ]. .. . λ−n+1
We then have that O − τ (Λ) is a strictly lower triangular Toeplitz matrix corresponding to the bounded analytic function f (z) =
∞ X
λ−i z i =
i=1
so that ||O − τ (Λ)|| ≤ sup | |z|j as a consequence of Lemma 8. Let vi denote row i of the matrix A, and compute the i, j-entry of AA∗ , which is < vi , vj >. With no loss of generality, assume that i ≤ j. When j − i is odd, we get Pi−1 2 < vi , vj > = k=1 fk fn−i+1 fn−j+1 P k−i+1 f + Pj−1 n−k fi−1 fk fn−j+1 k=i (−1) n 2 f f f − k=j n−k i−1 j−1 . We use the well known identity s X
fk2 = fs fs+1
k=1
on the first and last summands, and use Lemma 9 on the middle summand, obtaining < vi , vj > = fi−1 fi fn−i+1 fn−j+1 + fi−1 fn−j+1 (fj−1 fn−j − fi fn−i+1 ) − fi−1 fj−1 fn−j fn−j+1 = 0. When 0 ≤ j − i is even, we use Lemma Pi−1 2 < vi , vj > = k=1 fk fn−i+1 fn−j+1 + + = fi−1 fi fn−i+1 fn−j+1 + + = fn fi−1 fn−j+1 .
10 obtaining Pj−1 (−1)k−i+1 fn−k fi−1 fk fn−j+1 Pk=i n 2 k=j fn−k fi−1 fj−1 fi−1 fn−j+1 (fn − fj−1 fn−j − fi fn−i+1 ) fi−1 fj−1 fn−j fn−j+1
It follows that, with i ≤ j, the symmetric matrix AA∗ is determined by the values fn fi−1 fn−j+1 if j − i even < vi , vj >= 0 if j − i odd
12
Life is simplified a little if we delete the row and column of zeros, and let wi denote row i of the resulting (n − 1) × (n − 1) matrix. Now we have fn fi fn−j if j − i even < wi , wj >= 0 if j − i odd Being both nonnegative and symmetric, we have that the largest row sum dominates the norm (see [7]). Using the well known identities k−1 X
f2i+1 = f2k and
i=0
k X
f2i = f2k+1 − 1,
i=0
we see that each row sum is of the form fn fi X + fn fn−i Y, where X is either fn−i+1 or fn−i+1 − 1 and Y is either fi−1 or fi−1 − 1 (depending on whether i and n − i are even or odd). In every case, the row sums are dominated by fn fi fn−i+1 + fn fn−i fi−1 = fn2 , which completes the proof. Example 4
The well known identity n−1 X i=0
fi fi+1 =
if n even fn2 fn2 − 1 if n odd
lets one easily calculate the Frobenius norm of T , it is ( fq if n odd n+1 ||T ||F = . 2 fn+1 − 1 if n even The fact that the spectral norm of T grows rapidly, while all other singular values remain bounded by 1, explains why the Frobenius norm, always an upper bound of the spectral norm, quickly becomes a good approximation of the spectral norm for T . The previous theorem gives a lower bound. Indeed, since 2 s20 + (n − 1) ≥ tr (T T ∗ ) = fn+1 − δn , 13
with δn either zero or one, we have q 2 − n ≤ ||T || ≤ fn+1 . fn+1 When n = 12, mathematica gives the three values 232.974 ≤ 232.988 ≤ 233, which is already quite tight. The corresponding circulant matrix F (notation as in the beginning of section 4) has norm ||F || = fn+2 − 1 P (recall from the preliminary section that the norm equals the sum ni=0 fi ), || || √ 2||F || and for n = 12 we have the three ratios f||F ≤ ||F , computed ||T || ≤ n+1 fn+1 −n+1
in mathematica, as 1.6137339 ≤ 1.6138174 ≤ 1.6138974. By the time n gets to 24, all three values are within 5 decimal places of the golden ratio.
References [1] M. Bahsi and S. Solak, On the spectral norms of Hankel matrices with Fibonacci and Lucas numbers, Selcuk J. Appl. Math., 12 no. 1 (2011), 71-76. [2] D. Bozkurt and M. Akbulak, On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers, Hacet. J. Math. Stat., 37 (2)(2008), 89-95. [3] K. R. Davidson, Nest algebras, Pitman Research Notes in Mathematical Sciences, vol. 191, Longman Scientific and Technical, Harlow, (1988). [4] J. Dixon, B. Mathes, and D. Wheeler, An application of matricial Fibonacci identities to the computation of spectral norms, Rocky Mt. J. Math., to appear. [5] E. Dupree and B. Mathes, Singular values of k-Fibonacci and k-Lucas Hankel Matrices, Int. J. Contemp. Math. Sciences, to appear. [6] P. L. Duren, Theory of H p spaces. Dover, New York, 2000. 2nd. edition. 14
[7] R. A. Horn and C. A. Johnson, Matrix analysis, Cambridge University Press, Cambridge (1985). [8] A. Ipek, On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput., 217 (2011), 6011-6012. [9] D. Kalman and R. Mena, The Fibonacci numbers – exposed, Mathematics Magazine, vol. 76, no. 3 (2003) [10] I. Kra and S. Simanca, On circulant matrices, Notices of the AMS, vol. 59, no. 3 (2012) [11] G-Y. Lee, J-S. Kim, S-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. A special tribute to Calvin T. Long., Fibonacci Quart. 40 (2002), no. 3, 203-211. [12] N. K. Nikolskii, Treatise on the shift operator. Springer-Verlag, Berlin, 1986. Grundleheren der mathematischen Wissenschafte vol. 273. [13] W. Rudin, Real and complex analysis. McGraw-Hill, New York, 1986. [14] S. Shen, On the norms of Toeplitz matrices involving k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences, 7 (8) (2012), 363368. [15] S. Shen, J. Cen, On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences, 5 (12) (2010), 569-578. [16] S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160 (2005), 125-132. [17] E. W. Weisstein, CRC Encyclopedia of Mathematics, CRC Press, Chapman and Hall, vol. 1 (2009).
15
Abstract We prove a general theorem that gives tight bounds on the spectral norms of triangularly truncated k-Fibonacci and k-Lucas circulant matrices. The bounds are good enough to enable the calculation of the limit ||C|| , ||τ (C)|| as the dimension n approaches infinity, where τ (C) denotes the triangular truncation of C, and C is any n × n circulant matrix built using a sequence (si ) satisfying si = ksi−1 + si−2 . In particular, we have that this limit is equal to the golden ratio, if C is built using either the ordinary Fibonacci or Lucas sequence.
1
Introduction
The k-Fibonacci and k-Lucas sequences are second order recursive sequences satisfing fi−2 + kfi−1 = fi , for all integers i ≥ 2, the two sequences determined by their initial values; the k-Fibonacci sequence begins with f0 = 0 and f1 = 1, while the k-Lucas sequence begins f0 = 2 and f1 = k. We will follow the notation used in [9] and denote the set of all such recursive sequences by R(k, 1). When k = 1 we obtain the ordinary Fibonacci and Lucas sequence. There has recently 0 0
AMS 2000 Classification numbers: Primary: 15B36 Secondary: 65F35 Key Words: Triangular truncation, Fibonacci numbers, spectral norm.
1
been a flurry of interest in the norms and singular values of various matrices built using the Fibonacci and Lucas sequences, and their generalizations defined above ([1], [2], [4], [5], [8], [14], [15], [16]). In [15], the focus is on spectral norm estimates for r-circulant matrices whose entries are generated via the k-Fibonacci or k-Lucas sequence for positive k. The r-circulant matrices are of the form a0 a1 . . . an−2 an−1 ran−1 a0 a1 ... an−2 ran−2 ran−1 a0 ... an−3 A= , .. .. .. . . . . . . ra1
ra2
. . . ran−1
a0
thus if r = 1, then A is the n × n circulant matrix determined by the sequence (ai ) (see [10] for a nice expository article on circulant matrices), while A becomes the triangular truncation of a circulant matrix when r = 0. Here is the main result of [15] in the case of the k-Fibonacci sequence: Theorem 1 [15] Let A be as above, with ai = fi , the k-Fibonacci sequence. 1. If |r| ≥ 1, then r
fn fn−1 |r| − |r|n (fn + |r|fn−1 ) ≤ ||A|| ≤ . k 1 − k|r| − |r|2
2. If |r| < 1, then r |r|
fn + fn−1 − 1 fn fn−1 ≤ ||A|| ≤ . k k
When r = 1, the matrix A is circulant and, since its entries are nonnegative, n−1 X fn−1 + fn − 1 ||A|| = fi = , k i=0
(see the preliminary section). While (2) of Theorem 1 gives no non-trivial lower bound for r = 0, it does say that triangular truncation of a circulant matrix whose entries are generated by a k-Fibonacci is contractive. The aim of this note is to give, for any non-zero k, tight estimates on the spectral norm of the triangularly truncated A.
2
2
Preliminaries
The space R(k, 1) is a two dimensional (real or) complex vector space with a natural basis consisting of two sequences of the form (1, τ, τ 2 , τ 3 , ...), corresponding to the two distinct roots λ and µ of the polynomial τ 2 −kτ −1. These two roots satisfy 0 < |µ| < 1 < |λ|, and we refer to λ as the k-golden ratio. As it happens, −1 < µ < 0 and 1 < λ for positive k, while λ < −1 and 0 < µ < 1 when k is negative. One has λ + µ = k and λµ = −1, and writing the Fibonacci and Lucas sequences, (fi ) and (li ), in terms of this basis gives the Binet formulae: fi =
λi − µi and li = λi + µi , λ−µ
for all integers i. Almost all of this can be found in [9]. An n × n circulant matrix corresponding to a sequence (ai ) is the matrix a0 a1 . . . an−2 an−1 an−1 a0 a1 . . . an−2 A = an−2 an−1 a0 . . . an−3 , .. .. . . .. .. . . a1 a2 . . . an−1 a0 and the set of all such circulant matrices forms a commutative algebra C of normal matrices, which is simultaneously diagonalizable. If 1 n−1 E = √ uij i,j=0 , n with u a primitive nth -root of unity, then E is the unitary that implements the Discrete Fourier Transform, and it is also the unitary that diagonalizes circulant matrices, i.e. E ∗ AE is a diagonal matrix with diagonal entries n−1 X
ai v i ,
i=0
3
one entry for each nth -root of unity v, e.g. v = ui for i = 0, . . . , n − 1. We will refer to E as the DFT unitary. Most of this paragraph can be found in [10]. 1 The singular values of a matrix A are the eigenvalues of (A∗ A) 2 , and traditionally they are listed in descending order s0 ≥ s1 ≥ . . . ≥ sm > 0, with m + 1 equal to the rank of A. The spectral norm ||A|| is defined to be qP m 2 s0 , and the Frobenius norm ||A||F is defined to be i=0 si . Writing the entries [aij ] of the matrix A, we get X ||A||2F = |aij |2 , ij
and ||A|| = sup ||Ax||, ||x||≤1
where ||x|| denotes the Euclidean norm of x. Thus, the spectral norm is an operator norm. When A is a diagonal matrix, then the singular values are the absolute values of the diagonal entries, and more generally, when A is normal, the singular values are the absolute values of the eigenvalues of A. In particular, if A is an n × n circulant matrix generated from a sequence (ai ), then the spectral norm of A is the supremum over |
n−1 X
ai v i |,
i=0
as v varies through the nth roots of unity. When (ai ) is non-negative, then this supremum is attained with v = 1, so that ||A|| =
n−1 X
ai .
i=0
In case F is the circulant corresponding to the k-Fibonacci sequence with positive k, we get n−1 X fn−1 + fn − 1 ||F || = fi = , k i=0
which is the upper bound of the corresponding r-circulant matrix obtained in [15], for |r| ≤ 1. It appears that the authors were unaware of this at the 4
time they wrote their paper, while for the ordinary Fibonacci sequence, this fact is explicitly noted in [8]. The circulant matrices are examples of Toeplitz operators: they are constant along each upper left to lower right diagonal. An infinite matrix [aij ] acting boundedly on `2 is an analytic Toeplitz operator if it is Toeplitz and triangular. As such, the matrix is given by a sequence (ai ), and, assuming it is upper triangular, is of the form a0 a1 a2 . . . 0 a0 a1 a2 . . . A= 0 0 ... ... ... . .. . ... P i The corresponding function f (z) = ∞ i=0 ai z is a bounded analytic function in the open unit disc, and with ||A|| denoting the operator norm of A, one has ||A|| = sup |f (z)|. |z|≤1
If A is an n × n truncated circulant matrix corresponding to (ai ), and it P i a is known that f (z) = ∞ i=0 i z is a bounded analytic function in the unit disc, then A is a compression of A, hence ||A|| ≤ ||A|| = sup |f (z)|. |z|≤1
These facts can be found in any text that treats Hardy spaces: see [13], [12], and [6]. A matrix space, endowed with the spectral norm, lets us describe the distance from a matrix A to a set K as inf ||A − K||.
k∈K
If (si ) denotes the sequence of singular values of A, arranged in decreasing order, then si is exactly the distance from A to the set of matrices of rank less than or equal to i (for i = 0, . . . , rank(A)). This fact can be found in [7]. We intend to use this fact in the following form: given A, we find a matrix O of rank one with ||A − O|| ≤ r. We conclude that si ≤ r for all i ≥ 1.
5
3
Results
The first thought one might have, presented with evidence that truncating particular circulant matrices is contractive, is that it might be a property of all circulant matrices rather than the particular ones. As it happens, this is not the case. In fact, the rate of growth of the truncation norm, when restricted to the set of circulant n × n matrices, is the same as the rate of growth on all n × n matrices (see [3] for a proof that the norm of the triangular trunctation operator, defined on the set of all n × n matrices, grows on the order of log n). Theorem 2 Triangular truncation is unbounded when restricted to circulant matrices. In fact, there exists n × n circulant matrices A with n ||τ (A)|| 1 ≥ log . ||A|| π 2 Proof. Let E denote the DFT unitary. Let {e1 , . . . , en } denote the standard basis of Cn . Our plan is to show that there exists an n×n diagonal matrix D0 of norm one such that n 1 | < E ∗ τ (ED0 E ∗ )Ee1 , e1 > | ≥ log . π 2 Observe that for a circulant C = circ(c0 , . . . , cn−1 ), the northwest entry of the matrix E ∗ τ (C)E is n−1
1X < E τ (C)Ee1 , e1 >= (n − i)ci . n ∗
i=0
Apply this to the circulant C, where ci = primitive nth root of unity, i.e. the circulant
1 n
Pn−1 j=0
aj uij , and with u a
C = EDE ∗ , with D = diag(a0 , . . . an−1 ), to get n−1 n−1 n−1 n−1 X 1 X 1 X X ij | < E τ (EDE )Ee1 , e1 > | = | 2 (n−i) aj u | = | 2 aj (n−i)uij |, n n ∗
∗
i=0
j=0
j=0
i=0
from which we see that the supremum over all diagonal contractions D is attained, and equal to 6
n−1 n−1 1 X X | < E τ (ED0 E )Ee1 , e1 > | = 2 | (n − i)uij |. n ∗
∗
j=0
i=0
Pn−1
Let’s look at a typical sum i=0 (n − i)uij with 1 ≤ j: let v 6= 1 denote any nth root of unity. We massage the sum to appear like the derivative of a finite geometric series, obtaining |
n−1 X
i
(n − i)v | = |(n + 1)
i=0
When v = e
n−1 X i=0
2πki n
i
v −
n−1 X
i
(i + 1)v | = |
i=0
n−1 X
(i + 1)v i | =
i=0
n . |1 − v|
, we have
2kπ , n and realizing the distances |1 − v| come in conjugate pairs, we deduce Pb n−1 c n(n+1) 1 n ∗ ∗ 2 | < E τ (ED0 E )Ee1 , e1 > | ≥ n2 + 2 k=1 2πki 2 |1−e n | n−1 Pb c n2 ≥ n12 n(n+1) + 2 k=12 2kπ 2 Pb n−1 c 1 n ≥ π1 k=12 k1 ≥ π log 2 |1 − v| ≤
We let Λ denote the n × n circulant matrix generated by the sequence and M the n × n circulant matrix generated by (µi ), (with λ and µ the roots of τ 2 − kτ − 1 as above, and λ the k- golden ratio). The general n × n circulant generated from a sequence in R(k, 1) is then (λi ),
C = αΛ + βM, with α, β ∈ C. Its triangular truncation we denote by T = τ (C) = ατ (Λ) + βτ (M ). Assume from now on that k is any non-zero real number. Lemma 3 Let (si ) denote the sequence of singular values of τ (Λ). Then si ≤
1 |λ| − 1
for all i ≥ 1. 7
Proof.
Let O denote the rank one n × n matrix 1 λ−1 −2 O= λ [1 λ λ2 . . . λn−1 ]. .. . λ−n+1
We then have that O − τ (Λ) is a strictly lower triangular Toeplitz matrix corresponding to the bounded analytic function f (z) =
∞ X
λ−i z i =
i=1
so that ||O − τ (Λ)|| ≤ sup | |z|j as a consequence of Lemma 8. Let vi denote row i of the matrix A, and compute the i, j-entry of AA∗ , which is < vi , vj >. With no loss of generality, assume that i ≤ j. When j − i is odd, we get Pi−1 2 < vi , vj > = k=1 fk fn−i+1 fn−j+1 P k−i+1 f + Pj−1 n−k fi−1 fk fn−j+1 k=i (−1) n 2 f f f − k=j n−k i−1 j−1 . We use the well known identity s X
fk2 = fs fs+1
k=1
on the first and last summands, and use Lemma 9 on the middle summand, obtaining < vi , vj > = fi−1 fi fn−i+1 fn−j+1 + fi−1 fn−j+1 (fj−1 fn−j − fi fn−i+1 ) − fi−1 fj−1 fn−j fn−j+1 = 0. When 0 ≤ j − i is even, we use Lemma Pi−1 2 < vi , vj > = k=1 fk fn−i+1 fn−j+1 + + = fi−1 fi fn−i+1 fn−j+1 + + = fn fi−1 fn−j+1 .
10 obtaining Pj−1 (−1)k−i+1 fn−k fi−1 fk fn−j+1 Pk=i n 2 k=j fn−k fi−1 fj−1 fi−1 fn−j+1 (fn − fj−1 fn−j − fi fn−i+1 ) fi−1 fj−1 fn−j fn−j+1
It follows that, with i ≤ j, the symmetric matrix AA∗ is determined by the values fn fi−1 fn−j+1 if j − i even < vi , vj >= 0 if j − i odd
12
Life is simplified a little if we delete the row and column of zeros, and let wi denote row i of the resulting (n − 1) × (n − 1) matrix. Now we have fn fi fn−j if j − i even < wi , wj >= 0 if j − i odd Being both nonnegative and symmetric, we have that the largest row sum dominates the norm (see [7]). Using the well known identities k−1 X
f2i+1 = f2k and
i=0
k X
f2i = f2k+1 − 1,
i=0
we see that each row sum is of the form fn fi X + fn fn−i Y, where X is either fn−i+1 or fn−i+1 − 1 and Y is either fi−1 or fi−1 − 1 (depending on whether i and n − i are even or odd). In every case, the row sums are dominated by fn fi fn−i+1 + fn fn−i fi−1 = fn2 , which completes the proof. Example 4
The well known identity n−1 X i=0
fi fi+1 =
if n even fn2 fn2 − 1 if n odd
lets one easily calculate the Frobenius norm of T , it is ( fq if n odd n+1 ||T ||F = . 2 fn+1 − 1 if n even The fact that the spectral norm of T grows rapidly, while all other singular values remain bounded by 1, explains why the Frobenius norm, always an upper bound of the spectral norm, quickly becomes a good approximation of the spectral norm for T . The previous theorem gives a lower bound. Indeed, since 2 s20 + (n − 1) ≥ tr (T T ∗ ) = fn+1 − δn , 13
with δn either zero or one, we have q 2 − n ≤ ||T || ≤ fn+1 . fn+1 When n = 12, mathematica gives the three values 232.974 ≤ 232.988 ≤ 233, which is already quite tight. The corresponding circulant matrix F (notation as in the beginning of section 4) has norm ||F || = fn+2 − 1 P (recall from the preliminary section that the norm equals the sum ni=0 fi ), || || √ 2||F || and for n = 12 we have the three ratios f||F ≤ ||F , computed ||T || ≤ n+1 fn+1 −n+1
in mathematica, as 1.6137339 ≤ 1.6138174 ≤ 1.6138974. By the time n gets to 24, all three values are within 5 decimal places of the golden ratio.
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[7] R. A. Horn and C. A. Johnson, Matrix analysis, Cambridge University Press, Cambridge (1985). [8] A. Ipek, On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries, Appl. Math. Comput., 217 (2011), 6011-6012. [9] D. Kalman and R. Mena, The Fibonacci numbers – exposed, Mathematics Magazine, vol. 76, no. 3 (2003) [10] I. Kra and S. Simanca, On circulant matrices, Notices of the AMS, vol. 59, no. 3 (2012) [11] G-Y. Lee, J-S. Kim, S-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices. A special tribute to Calvin T. Long., Fibonacci Quart. 40 (2002), no. 3, 203-211. [12] N. K. Nikolskii, Treatise on the shift operator. Springer-Verlag, Berlin, 1986. Grundleheren der mathematischen Wissenschafte vol. 273. [13] W. Rudin, Real and complex analysis. McGraw-Hill, New York, 1986. [14] S. Shen, On the norms of Toeplitz matrices involving k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences, 7 (8) (2012), 363368. [15] S. Shen, J. Cen, On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sciences, 5 (12) (2010), 569-578. [16] S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput., 160 (2005), 125-132. [17] E. W. Weisstein, CRC Encyclopedia of Mathematics, CRC Press, Chapman and Hall, vol. 1 (2009).
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