- Email: [email protected]
Powers of real persymmetric anti-tridiagonal matrices with constant anti-diagonals
Applied Mathematics and Computation 206 (2008) 919–924
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Powers of real persymmetric anti-tridiagonal matrices with constant anti-diagonals Jesús Gutiérrez-Gutiérrez CEIT and Tecnun (University of Navarra), Manuel de Lardizábal 15, E-20018 San Sebastián, Spain
a r t i c l e
i n f o
a b s t r a c t In this paper, we present an eigenvalue decomposition with an orthogonal eigenvector matrix for any real persymmetric anti-tridiagonal matrix with constant anti-diagonals. Furthermore, we derive a general expression for the entries of the powers of these anti-tridiagonal matrices in terms of the Chebyshev polynomials of the second kind. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Anti-tridiagonal matrices Eigenvalues Eigenvectors Chebyshev polynomials
1. Introduction In the present paper, we study the entries of positive integer powers of an n n real persymmetric anti-tridiagonal matrix with constant anti-diagonals
0
0 .. . .. .
B B B B B B Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ :¼ B B B0 B B B @ a1 a0
0
a1
a0
1
C a1 C C C C q a1 a0 a1 0 C C; . C q q q q .. C C C .. C q q q . A a1
q
a1
a0
0
0
where a0 ; a1 2 R. In the next section, we present an eigenvalue decomposition of the Hankel (constant anti-diagonals) matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the eigenvector matrix is orthogonal. In the last section, based on this decomposition we introduce Corollary 6 that provides a general expression for the entries of Bqn for all q; n 2 N. This novel expression is an extension of the one obtained by Rimas for the powers of the matrix antitridiagn ð1; 0; 1Þ with n 2 N (see [1] for the odd case and [2] for the even case).
2. An eigenvalue decomposition of Bn We begin this section by reviewing a theorem regarding an n n Hermitian Toeplitz (constant diagonals) tridiagonal matrix
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.10.003
920
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
0
a0
B Ba B 1 B B B0 An ¼ tridiagn ða1 ; a0 ; a1 Þ :¼ B B . B .. B B . B . @ . 0
1 0 .. .. C a0 a1 . . C C C .. .. C a1 a0 a1 . . C C; C .. .. .. .. . . . . 0C C C .. .. .. C . . . a1 A 0 a1 a0 a1
0
where a0 2 R. This result was proved in [3] and provides an eigenvalue decomposition of An with unitary eigenvector matrix. Theorem 1. Let a0 2 R, a1 –0 and n 2 N. Then W n Dn W n is an eigenvalue decomposition of An ¼ tridiagn ða1 ; a0 ; a1 Þ, where the entries of the unitary eigenvector matrix W n are
½W n j;k ¼
rffiffiffiffiffiffiffiffiffiffiffiffi j 2 a1 jkp sin ; n þ 1 ja1 j nþ1
1 6 j; k 6 n;
ð1Þ
Dn ¼ diagða1 ; . . . ; an Þ with
aj ¼ a0 þ 2ja1 j cos
jp ; nþ1
1 6 j 6 n;
ð2Þ
and denotes conjugate transpose ði:e:; W n ¼ W n > Þ. The next trivial result relates the matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ with An ¼ tridiagn ða1 ; a0 ; a1 Þ and with the n n backward identity (see e.g. [4])
1 0 0 1 C B. B .. q 1 0C C B C B .. C B .. ¼ B . q q q . C; C B B .. C C B .A @0 q q 1 0 0 0
J n ¼ ðdjþk;nþ1 Þnj;k¼1
where d is the Kronecker delta. Lemma 2. Let a0 ; a1 2 R and n 2 N. Then
Bn ¼ J n An with Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ and An ¼ tridiagn ða1 ; a0 ; a1 Þ. In the following lemma we present an eigenvalue decomposition of J n , where the eigenvector matrix is orthogonal. Lemma 3. Let a1 2 R n f0g and n 2 N. Then
J n ¼ W n diagðl1 ; . . . ; ln ÞW >n ;
ð3Þ
where
lj ¼ ð1Þjþ1
a1 ja1 j
nþ1 ;
1 6 j 6 n;
ð4Þ
and the entries of the real orthogonal matrix W n are given by (1). Proof. We will split the proof into two steps: Step 1: In this step we will show that
8 0 > > > > > >
if n þ 1 þ m is odd;
if n þ 1 þ m 2 X and jn þ 1 mj 2 X; n X mhp ¼ n1 ð1Þh cos if n þ 1 þ m 2 X and jn þ 1 mj 2 2N n X; 2 nþ1 > > h¼1 n1 > > if n þ 1 þ m 2 2N n X and jn þ 1 mj 2 X; > > 2 : 1 if n þ 1 þ m 2 2N n X and jn þ 1 mj 2 2N n X; for every n 2 N and m 2 N
S
f0g, where X ¼ f2kðn þ 1Þ : k 2 N
S
f0gg.
ð5Þ
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
921
We have
" # n n n n X mhp X ðn þ 1Þhp mhp 1 X ðn þ 1 þ mÞhp X jn þ 1 mjhp : ð1Þh cos cos cos cos ¼ cos ¼ þ n þ 1 h¼1 nþ1 n þ 1 2 h¼1 nþ1 nþ1 h¼1 h¼1
ð6Þ
Observe that equality (5) can be deduced from expression (6), by using the fact that n þ 1 þ m is odd iff jn þ 1 mj is odd and by taking into account the following equality n X h¼1
8 if m is odd; >0 mhp < cos ¼ n if m 2 X; nþ1 > : 1 if m 2 2N n X;
ð7Þ
S for every n 2 N and m 2 N f0g. Finally, it should be mentioned that the proof of equality (7) is similar to the proof of expression (2) in [3]. Step 2: In the second step based on expression (5) we will prove the assertion of the lemma, i.e., expression (3). Notice that (3) is equivalent to
W n Dn W >n ¼
a1 ja1 j
nþ1 Jn
with Dn ¼ diagð1; 1; 1; . . . ; ð1Þn Þ. We have
½W n Dn W >n j;k ¼
n n n n X X X X ½W n j;h ½Dn W >n h;k ¼ ½W n j;h ½Dn h;l ½W >n l;k ¼ ð1Þh ½W n j;h ½W n k;h h¼1
h¼1
l¼1
h¼1
" # n n n 2c X jhp khp c X jj kjhp X ðj þ kÞhp h h h ¼ ð1Þ sin ð1Þ cos ð1Þ cos sin ¼ n þ 1 h¼1 nþ1 n þ 1 n þ 1 h¼1 nþ1 nþ1 h¼1
jþk with c ¼ jaa11 j . On the one hand, if n þ 1 þ ðj þ kÞ is odd then n þ 1 þ jj kj is also odd and consequently by using (5) we obtain that n n X jj kjhp X ðj þ kÞhp ð1Þh cos ð1Þh cos ¼ ¼0 nþ1 nþ1 h¼1 h¼1
c and therefore, ½W n Dn W > n j;k ¼ nþ1 ½0 0 ¼ 0. On the other hand, if n þ 1 þ ðj þ kÞ is even then jn þ 1 ðj þ kÞj, n þ 1 þ jj kj and n þ 1 jj kj are also even. Since n þ 1 6 n þ 1 þ jj kj 6 2n and 2 6 n þ 1 jj kj 6 n þ 1 then n þ 1 þ jj kj and jn þ 1 jj kjj belong to 2N n X. Thus, from (5) n X jj kjhp ¼ 1: ð1Þh cos nþ1 h¼1
Since n þ 3 6 n þ 1 þ ðj þ kÞ 6 3n þ 1 and 0 6 jn þ 1 ðj þ kÞj 6 n 1 then
n þ 1 þ ðj þ kÞ 2 X () n þ 1 þ ðj þ kÞ ¼ 2ðn þ 1Þ () j þ k ¼ n þ 1 () jn þ 1 ðj þ kÞj ¼ 0 () jn þ 1 ðj þ kÞj 2 X: Therefore, if n þ 1 þ ðj þ kÞ 2 2N n X then from (5) n X ðj þ kÞhp ð1Þh cos ¼ 1 nþ1 h¼1
c and consequently, ½W n Dn W > n j;k ¼ nþ1 ½1 ð1Þ ¼ 0. If n þ 1 þ ðj þ kÞ 2 X (that is, j þ k ¼ n þ 1) by using (5) we obtain that n X ðj þ kÞhp ð1Þh cos ¼n nþ1 h¼1
c and therefore, ½W n Dn W > n j;k ¼ nþ1 ½1 n ¼ c. Summarizing,
½W n Dn W >n j;k ¼
c if j þ k ¼ n þ 1; 0
nþ1 a1 i.e., W n Dn W > Jn . n ¼ ja1 j
in other case;
nþ1 a1 djþk;nþ1 ; ¼ ja1 j
h
By combining Theorem Lemmas 1–3 we directly obtain an eigenvalue decomposition of the matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the eigenvector matrix is orthogonal.
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J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
Theorem 4. Let a0 2 R, a1 2 R n f0g and n 2 N. Then W n diagðs1 ; . . . ; sn ÞW > is an eigenvalue decomposition of n Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the entries of the real orthogonal eigenvector matrix W n are given by (1) and the eigenvalues by
sj ¼ ð1Þjþ1
a1 ja1 j
nþ1
aj ; 1 6 j 6 n;
with the aj defined by expression (2). 3. General expression for the entries of Bqn Let U m ðxÞ be the mth degree Chebyshev polynomial of the second kind, with m 2 N [ f0g
U m ðxÞ ¼
sin½ðm þ 1Þ arccos x ; sin arccos x
1 < x < 1:
We now review a theorem regarding powers of An ¼ tridiagn ða1 ; a0 ; a1 Þ. This result was proved in [3] and gives a general expression for the entries of Aqn for all q; n 2 N in terms of the Chebyshev polynomials of the second kind. hp Theorem 5. Consider a0 2 R, a1 –0 and n 2 N. Let An ¼ tridiagn ða1 ; a0 ; a1 Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
jk h 1 a1 2ð1 þ ð1Þnþ1 Þaq0 U j1 ð0ÞU k1 ð0Þ 2n þ 2 ja1 j bn2c X knhþ1 knhþ1 þ ð4 k2nhþ1 ÞU j1 U k1 ½ða0 þ ja1 jknhþ1 Þq þ ð1Þjþk ða0 ja1 jknhþ1 Þq 2 2 h¼1
½Aqn j;k ¼
for all q 2 N and 1 6 j; k 6 n, where bxc denotes the largest integer less than or equal to x. Consider a0 2 R and a1 2 R n f0g. We now proceed to obtain a general expression for the entries of Bqn ¼ ðantitridiagn ða1 ; a0 ; a1 ÞÞq for all q; n 2 N in terms of the Chebyshev polynomials of the second kind. From Theorem 4 we have
q Bqn ¼ W n diagðs1 ; . . . ; sn ÞW >n ¼ W n diagðsq1 ; . . . ; sqn ÞW >n ( ( W n diagðaq1 ; . . . ; aqn ÞW >n if q is even; W n diagðaq1 ; . . . ; aqn ÞW >n ¼ ¼ q q > W n diagðl1 a1 ; . . . ; ln an ÞW n if q is odd; W n diagðl1 ; . . . ; ln ÞW >n W n diagðaq1 ; . . . ; aqn ÞW >n where the
if q is odd;
lj are given by (4). Let An ¼ tridiagn ða1 ; a0 ; a1 Þ, then by using Theorem 1 and Lemma 3 we obtain that (
Bqn
if q is even;
Aqn J n Aqn
¼
if q is even; if q is odd:
ð8Þ
From (8) we have
½Bqn j;k ¼
8 q > < ½An j;k
if q is even;
q > : ½Jn An j;k ¼
n P
½J n j;h ½Aqn h;k ¼ ½Aqn nþ1j;k
if q is odd;
h¼1
for all 1 6 j; k 6 n. Consequently, we can obtain the general expression for the entries of Bqn as a corollary of Theorem 5. hp Corollary 6. Consider a0 2 R, a1 2 R n f0g and n 2 N. Let Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
^jk h 1 a1 2ð1 þ ð1Þnþ1 Þaq0 U^j1 ð0ÞU k1 ð0Þ 2n þ 2 ja1 j b2nc X ^ knhþ1 knhþ1 U k1 ½ða0 þ ja1 jknhþ1 Þq þ ð1Þjþk ða0 ja1 jknhþ1 Þq ð4 k2nhþ1 ÞU^j1 þ 2 2 h¼1
½Bqn j;k ¼
for all q 2 N and 1 6 j; k 6 n, where
^j ¼
j
if q is even;
n þ 1 j if q is odd:
ð9Þ
From Corollary 6 we can now easily deduce the expression given by Rimas for the entries of the powers of the matrix antitridiagn ð1; 0; 1Þ with odd n [1].
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
923
hp Theorem 7. Consider an odd natural number n. Let Bn ¼ antitridiagn ð1; 0; 1Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
2
1 þ ð1Þqþjþk X kh kh U nk ¼ bj;k ðhÞkqh 4 k2h U nj 2n þ 2 2 2 h¼1 n1
½Bqn j;k
for all q 2 N and 1 6 j; k 6 n, where
( bj;k ðhÞ ¼
1
if j þ k is even; h1
ð1Þ
if j þ k is odd:
Proof. From Corollary 6
^ 2
1 þ ð1Þqþjþk X knhþ1 knhþ1 ¼ kqnhþ1 4 k2nhþ1 U^j1 U k1 2n þ 2 2 2 h¼1 n1
½Bqn j;k
p hp with ^j given by (9). Since cos nþ1 ¼ cos ðnþ1hÞ , 1 6 h 6 n, then nþ1
knhþ1 ¼ kh ;
1 6 h 6 n:
Consequently, by using
U m ðxÞ ¼ ð1Þm U m ðxÞ;
m 2 N [ f0g
(i.e., the mth degree Chebyshev polynomial of the second kind is an even or odd function, when m is even or odd respectively), we obtain that
qþ^jþk
½Bqn j;k ¼
1 þ ð1Þ 2n þ 2
8 n1 2 >
> 1þð1Þqþjþk P q > if q is even; kh 4 k2h U j1 k2h U k1 k2h > < 2nþ2 X q
kh kh h¼1 kh 4 k2h U^j1 U k1 ¼ n1 > 2 2 2 >
qþnþ1jþk P h¼1 > > kqh 4 k2h U nj k2h U k1 k2h if q is odd: : 1þð1Þ 2nþ2 n1 2
h¼1
Notice that
U m1
ðnmþ1Þhp p mhp sin mh kh kh hm cosðhpÞ sin nþ1 hm sin nþ1 nþ1 ¼ ð1Þ ¼ ð1Þ ¼ ð1Þm1 U m1 ¼ ð1Þm1 hp hp hp 2 2 sin nþ1 sin nþ1 sin nþ1 kh kh ¼ ð1Þhm U nm ¼ ð1Þhn U nm 2 2
with m ¼ j; k and 1 6 h 6 n. Therefore,
½Bqn j;k ¼
8 n1 2 >
qþjþk P > > 1þð1Þ kqh 4 k2h U nj k2h U nk k2h > < 2nþ2
if q is even;
h¼1
n1 > 2 >
qþnþ1jþk P > > ð1Þhn kqh 4 k2h U nj k2h U nk k2h if q is odd; : 1þð1Þ 2nþ2
h¼1
¼
¼
8 n1 2 >
> 1þð1Þqþjþk P q > kh 4 k2h U nj k2h U nk k2h > < 2nþ2
if q is even;
h¼1
n1 > 2 >
qþjþk P > > ð1Þh1 kqh 4 k2h U nj k2h U nk k2h if q is odd; : 1þð1Þ 2nþ2 h¼1 8 n1 2 >
qþjþk P > 1þð1Þ > kqh 4 k2h U nj k2h U nk k2h if j þ k is even; > < 2nþ2
h¼1
n1 > 2 >
qþjþk P > > ð1Þh1 kqh 4 k2h U nj k2h U nk k2h if j þ k is odd: : 1þð1Þ 2nþ2
h¼1
This ends the proof. h The expression given by Rimas for the entries of the powers of antitridiagn ð1; 0; 1Þ when n is even [2] can also be obtained from Corollary 6 in a similar way.
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J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
Acknowledgements I would like to thank Dr. Adam Podhorski and the anonymous reviewer for their helpful comments and criticism. This work was partially supported by the Spanish Ministy of Education and Science, and by the European Regional Development Fund through the MultiMIMO Project TEC2007-68020-C04-03 and the program CONSOLIDER-INGENIO 2010 CSD2008-00010 COMONSENS. References [1] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric anti-tridiagonal matrices of odd order, Appl. Math. Comput. 203 (2008) 573–581. [2] J. Rimas, On computing of arbitrary integer powers for one type of symmetric anti-tridiagonal matrices of even order, Appl. Math. Comput. 204 (2008) 288–298. [3] J. Gutiérrez-Gutiérrez, Positive integer powers of certain tridiagonal matrices, Appl. Math. Comput. 202 (2008) 133–140. [4] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1990.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Powers of real persymmetric anti-tridiagonal matrices with constant anti-diagonals Jesús Gutiérrez-Gutiérrez CEIT and Tecnun (University of Navarra), Manuel de Lardizábal 15, E-20018 San Sebastián, Spain
a r t i c l e
i n f o
a b s t r a c t In this paper, we present an eigenvalue decomposition with an orthogonal eigenvector matrix for any real persymmetric anti-tridiagonal matrix with constant anti-diagonals. Furthermore, we derive a general expression for the entries of the powers of these anti-tridiagonal matrices in terms of the Chebyshev polynomials of the second kind. Ó 2008 Elsevier Inc. All rights reserved.
Keywords: Anti-tridiagonal matrices Eigenvalues Eigenvectors Chebyshev polynomials
1. Introduction In the present paper, we study the entries of positive integer powers of an n n real persymmetric anti-tridiagonal matrix with constant anti-diagonals
0
0 .. . .. .
B B B B B B Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ :¼ B B B0 B B B @ a1 a0
0
a1
a0
1
C a1 C C C C q a1 a0 a1 0 C C; . C q q q q .. C C C .. C q q q . A a1
q
a1
a0
0
0
where a0 ; a1 2 R. In the next section, we present an eigenvalue decomposition of the Hankel (constant anti-diagonals) matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the eigenvector matrix is orthogonal. In the last section, based on this decomposition we introduce Corollary 6 that provides a general expression for the entries of Bqn for all q; n 2 N. This novel expression is an extension of the one obtained by Rimas for the powers of the matrix antitridiagn ð1; 0; 1Þ with n 2 N (see [1] for the odd case and [2] for the even case).
2. An eigenvalue decomposition of Bn We begin this section by reviewing a theorem regarding an n n Hermitian Toeplitz (constant diagonals) tridiagonal matrix
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.10.003
920
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
0
a0
B Ba B 1 B B B0 An ¼ tridiagn ða1 ; a0 ; a1 Þ :¼ B B . B .. B B . B . @ . 0
1 0 .. .. C a0 a1 . . C C C .. .. C a1 a0 a1 . . C C; C .. .. .. .. . . . . 0C C C .. .. .. C . . . a1 A 0 a1 a0 a1
0
where a0 2 R. This result was proved in [3] and provides an eigenvalue decomposition of An with unitary eigenvector matrix. Theorem 1. Let a0 2 R, a1 –0 and n 2 N. Then W n Dn W n is an eigenvalue decomposition of An ¼ tridiagn ða1 ; a0 ; a1 Þ, where the entries of the unitary eigenvector matrix W n are
½W n j;k ¼
rffiffiffiffiffiffiffiffiffiffiffiffi j 2 a1 jkp sin ; n þ 1 ja1 j nþ1
1 6 j; k 6 n;
ð1Þ
Dn ¼ diagða1 ; . . . ; an Þ with
aj ¼ a0 þ 2ja1 j cos
jp ; nþ1
1 6 j 6 n;
ð2Þ
and denotes conjugate transpose ði:e:; W n ¼ W n > Þ. The next trivial result relates the matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ with An ¼ tridiagn ða1 ; a0 ; a1 Þ and with the n n backward identity (see e.g. [4])
1 0 0 1 C B. B .. q 1 0C C B C B .. C B .. ¼ B . q q q . C; C B B .. C C B .A @0 q q 1 0 0 0
J n ¼ ðdjþk;nþ1 Þnj;k¼1
where d is the Kronecker delta. Lemma 2. Let a0 ; a1 2 R and n 2 N. Then
Bn ¼ J n An with Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ and An ¼ tridiagn ða1 ; a0 ; a1 Þ. In the following lemma we present an eigenvalue decomposition of J n , where the eigenvector matrix is orthogonal. Lemma 3. Let a1 2 R n f0g and n 2 N. Then
J n ¼ W n diagðl1 ; . . . ; ln ÞW >n ;
ð3Þ
where
lj ¼ ð1Þjþ1
a1 ja1 j
nþ1 ;
1 6 j 6 n;
ð4Þ
and the entries of the real orthogonal matrix W n are given by (1). Proof. We will split the proof into two steps: Step 1: In this step we will show that
8 0 > > > > > >
if n þ 1 þ m is odd;
if n þ 1 þ m 2 X and jn þ 1 mj 2 X; n X mhp ¼ n1 ð1Þh cos if n þ 1 þ m 2 X and jn þ 1 mj 2 2N n X; 2 nþ1 > > h¼1 n1 > > if n þ 1 þ m 2 2N n X and jn þ 1 mj 2 X; > > 2 : 1 if n þ 1 þ m 2 2N n X and jn þ 1 mj 2 2N n X; for every n 2 N and m 2 N
S
f0g, where X ¼ f2kðn þ 1Þ : k 2 N
S
f0gg.
ð5Þ
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
921
We have
" # n n n n X mhp X ðn þ 1Þhp mhp 1 X ðn þ 1 þ mÞhp X jn þ 1 mjhp : ð1Þh cos cos cos cos ¼ cos ¼ þ n þ 1 h¼1 nþ1 n þ 1 2 h¼1 nþ1 nþ1 h¼1 h¼1
ð6Þ
Observe that equality (5) can be deduced from expression (6), by using the fact that n þ 1 þ m is odd iff jn þ 1 mj is odd and by taking into account the following equality n X h¼1
8 if m is odd; >0 mhp < cos ¼ n if m 2 X; nþ1 > : 1 if m 2 2N n X;
ð7Þ
S for every n 2 N and m 2 N f0g. Finally, it should be mentioned that the proof of equality (7) is similar to the proof of expression (2) in [3]. Step 2: In the second step based on expression (5) we will prove the assertion of the lemma, i.e., expression (3). Notice that (3) is equivalent to
W n Dn W >n ¼
a1 ja1 j
nþ1 Jn
with Dn ¼ diagð1; 1; 1; . . . ; ð1Þn Þ. We have
½W n Dn W >n j;k ¼
n n n n X X X X ½W n j;h ½Dn W >n h;k ¼ ½W n j;h ½Dn h;l ½W >n l;k ¼ ð1Þh ½W n j;h ½W n k;h h¼1
h¼1
l¼1
h¼1
" # n n n 2c X jhp khp c X jj kjhp X ðj þ kÞhp h h h ¼ ð1Þ sin ð1Þ cos ð1Þ cos sin ¼ n þ 1 h¼1 nþ1 n þ 1 n þ 1 h¼1 nþ1 nþ1 h¼1
jþk with c ¼ jaa11 j . On the one hand, if n þ 1 þ ðj þ kÞ is odd then n þ 1 þ jj kj is also odd and consequently by using (5) we obtain that n n X jj kjhp X ðj þ kÞhp ð1Þh cos ð1Þh cos ¼ ¼0 nþ1 nþ1 h¼1 h¼1
c and therefore, ½W n Dn W > n j;k ¼ nþ1 ½0 0 ¼ 0. On the other hand, if n þ 1 þ ðj þ kÞ is even then jn þ 1 ðj þ kÞj, n þ 1 þ jj kj and n þ 1 jj kj are also even. Since n þ 1 6 n þ 1 þ jj kj 6 2n and 2 6 n þ 1 jj kj 6 n þ 1 then n þ 1 þ jj kj and jn þ 1 jj kjj belong to 2N n X. Thus, from (5) n X jj kjhp ¼ 1: ð1Þh cos nþ1 h¼1
Since n þ 3 6 n þ 1 þ ðj þ kÞ 6 3n þ 1 and 0 6 jn þ 1 ðj þ kÞj 6 n 1 then
n þ 1 þ ðj þ kÞ 2 X () n þ 1 þ ðj þ kÞ ¼ 2ðn þ 1Þ () j þ k ¼ n þ 1 () jn þ 1 ðj þ kÞj ¼ 0 () jn þ 1 ðj þ kÞj 2 X: Therefore, if n þ 1 þ ðj þ kÞ 2 2N n X then from (5) n X ðj þ kÞhp ð1Þh cos ¼ 1 nþ1 h¼1
c and consequently, ½W n Dn W > n j;k ¼ nþ1 ½1 ð1Þ ¼ 0. If n þ 1 þ ðj þ kÞ 2 X (that is, j þ k ¼ n þ 1) by using (5) we obtain that n X ðj þ kÞhp ð1Þh cos ¼n nþ1 h¼1
c and therefore, ½W n Dn W > n j;k ¼ nþ1 ½1 n ¼ c. Summarizing,
½W n Dn W >n j;k ¼
c if j þ k ¼ n þ 1; 0
nþ1 a1 i.e., W n Dn W > Jn . n ¼ ja1 j
in other case;
nþ1 a1 djþk;nþ1 ; ¼ ja1 j
h
By combining Theorem Lemmas 1–3 we directly obtain an eigenvalue decomposition of the matrix Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the eigenvector matrix is orthogonal.
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Theorem 4. Let a0 2 R, a1 2 R n f0g and n 2 N. Then W n diagðs1 ; . . . ; sn ÞW > is an eigenvalue decomposition of n Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ, where the entries of the real orthogonal eigenvector matrix W n are given by (1) and the eigenvalues by
sj ¼ ð1Þjþ1
a1 ja1 j
nþ1
aj ; 1 6 j 6 n;
with the aj defined by expression (2). 3. General expression for the entries of Bqn Let U m ðxÞ be the mth degree Chebyshev polynomial of the second kind, with m 2 N [ f0g
U m ðxÞ ¼
sin½ðm þ 1Þ arccos x ; sin arccos x
1 < x < 1:
We now review a theorem regarding powers of An ¼ tridiagn ða1 ; a0 ; a1 Þ. This result was proved in [3] and gives a general expression for the entries of Aqn for all q; n 2 N in terms of the Chebyshev polynomials of the second kind. hp Theorem 5. Consider a0 2 R, a1 –0 and n 2 N. Let An ¼ tridiagn ða1 ; a0 ; a1 Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
jk h 1 a1 2ð1 þ ð1Þnþ1 Þaq0 U j1 ð0ÞU k1 ð0Þ 2n þ 2 ja1 j bn2c X knhþ1 knhþ1 þ ð4 k2nhþ1 ÞU j1 U k1 ½ða0 þ ja1 jknhþ1 Þq þ ð1Þjþk ða0 ja1 jknhþ1 Þq 2 2 h¼1
½Aqn j;k ¼
for all q 2 N and 1 6 j; k 6 n, where bxc denotes the largest integer less than or equal to x. Consider a0 2 R and a1 2 R n f0g. We now proceed to obtain a general expression for the entries of Bqn ¼ ðantitridiagn ða1 ; a0 ; a1 ÞÞq for all q; n 2 N in terms of the Chebyshev polynomials of the second kind. From Theorem 4 we have
q Bqn ¼ W n diagðs1 ; . . . ; sn ÞW >n ¼ W n diagðsq1 ; . . . ; sqn ÞW >n ( ( W n diagðaq1 ; . . . ; aqn ÞW >n if q is even; W n diagðaq1 ; . . . ; aqn ÞW >n ¼ ¼ q q > W n diagðl1 a1 ; . . . ; ln an ÞW n if q is odd; W n diagðl1 ; . . . ; ln ÞW >n W n diagðaq1 ; . . . ; aqn ÞW >n where the
if q is odd;
lj are given by (4). Let An ¼ tridiagn ða1 ; a0 ; a1 Þ, then by using Theorem 1 and Lemma 3 we obtain that (
Bqn
if q is even;
Aqn J n Aqn
¼
if q is even; if q is odd:
ð8Þ
From (8) we have
½Bqn j;k ¼
8 q > < ½An j;k
if q is even;
q > : ½Jn An j;k ¼
n P
½J n j;h ½Aqn h;k ¼ ½Aqn nþ1j;k
if q is odd;
h¼1
for all 1 6 j; k 6 n. Consequently, we can obtain the general expression for the entries of Bqn as a corollary of Theorem 5. hp Corollary 6. Consider a0 2 R, a1 2 R n f0g and n 2 N. Let Bn ¼ antitridiagn ða1 ; a0 ; a1 Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
^jk h 1 a1 2ð1 þ ð1Þnþ1 Þaq0 U^j1 ð0ÞU k1 ð0Þ 2n þ 2 ja1 j b2nc X ^ knhþ1 knhþ1 U k1 ½ða0 þ ja1 jknhþ1 Þq þ ð1Þjþk ða0 ja1 jknhþ1 Þq ð4 k2nhþ1 ÞU^j1 þ 2 2 h¼1
½Bqn j;k ¼
for all q 2 N and 1 6 j; k 6 n, where
^j ¼
j
if q is even;
n þ 1 j if q is odd:
ð9Þ
From Corollary 6 we can now easily deduce the expression given by Rimas for the entries of the powers of the matrix antitridiagn ð1; 0; 1Þ with odd n [1].
J. Gutiérrez-Gutiérrez / Applied Mathematics and Computation 206 (2008) 919–924
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hp Theorem 7. Consider an odd natural number n. Let Bn ¼ antitridiagn ð1; 0; 1Þ and kh ¼ 2 cos nþ1 for every 1 6 h 6 n. Then
2
1 þ ð1Þqþjþk X kh kh U nk ¼ bj;k ðhÞkqh 4 k2h U nj 2n þ 2 2 2 h¼1 n1
½Bqn j;k
for all q 2 N and 1 6 j; k 6 n, where
( bj;k ðhÞ ¼
1
if j þ k is even; h1
ð1Þ
if j þ k is odd:
Proof. From Corollary 6
^ 2
1 þ ð1Þqþjþk X knhþ1 knhþ1 ¼ kqnhþ1 4 k2nhþ1 U^j1 U k1 2n þ 2 2 2 h¼1 n1
½Bqn j;k
p hp with ^j given by (9). Since cos nþ1 ¼ cos ðnþ1hÞ , 1 6 h 6 n, then nþ1
knhþ1 ¼ kh ;
1 6 h 6 n:
Consequently, by using
U m ðxÞ ¼ ð1Þm U m ðxÞ;
m 2 N [ f0g
(i.e., the mth degree Chebyshev polynomial of the second kind is an even or odd function, when m is even or odd respectively), we obtain that
qþ^jþk
½Bqn j;k ¼
1 þ ð1Þ 2n þ 2
8 n1 2 >
> 1þð1Þqþjþk P q > if q is even; kh 4 k2h U j1 k2h U k1 k2h > < 2nþ2 X q
kh kh h¼1 kh 4 k2h U^j1 U k1 ¼ n1 > 2 2 2 >
qþnþ1jþk P h¼1 > > kqh 4 k2h U nj k2h U k1 k2h if q is odd: : 1þð1Þ 2nþ2 n1 2
h¼1
Notice that
U m1
ðnmþ1Þhp p mhp sin mh kh kh hm cosðhpÞ sin nþ1 hm sin nþ1 nþ1 ¼ ð1Þ ¼ ð1Þ ¼ ð1Þm1 U m1 ¼ ð1Þm1 hp hp hp 2 2 sin nþ1 sin nþ1 sin nþ1 kh kh ¼ ð1Þhm U nm ¼ ð1Þhn U nm 2 2
with m ¼ j; k and 1 6 h 6 n. Therefore,
½Bqn j;k ¼
8 n1 2 >
qþjþk P > > 1þð1Þ kqh 4 k2h U nj k2h U nk k2h > < 2nþ2
if q is even;
h¼1
n1 > 2 >
qþnþ1jþk P > > ð1Þhn kqh 4 k2h U nj k2h U nk k2h if q is odd; : 1þð1Þ 2nþ2
h¼1
¼
¼
8 n1 2 >
> 1þð1Þqþjþk P q > kh 4 k2h U nj k2h U nk k2h > < 2nþ2
if q is even;
h¼1
n1 > 2 >
qþjþk P > > ð1Þh1 kqh 4 k2h U nj k2h U nk k2h if q is odd; : 1þð1Þ 2nþ2 h¼1 8 n1 2 >
qþjþk P > 1þð1Þ > kqh 4 k2h U nj k2h U nk k2h if j þ k is even; > < 2nþ2
h¼1
n1 > 2 >
qþjþk P > > ð1Þh1 kqh 4 k2h U nj k2h U nk k2h if j þ k is odd: : 1þð1Þ 2nþ2
h¼1
This ends the proof. h The expression given by Rimas for the entries of the powers of antitridiagn ð1; 0; 1Þ when n is even [2] can also be obtained from Corollary 6 in a similar way.
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Acknowledgements I would like to thank Dr. Adam Podhorski and the anonymous reviewer for their helpful comments and criticism. This work was partially supported by the Spanish Ministy of Education and Science, and by the European Regional Development Fund through the MultiMIMO Project TEC2007-68020-C04-03 and the program CONSOLIDER-INGENIO 2010 CSD2008-00010 COMONSENS. References [1] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric anti-tridiagonal matrices of odd order, Appl. Math. Comput. 203 (2008) 573–581. [2] J. Rimas, On computing of arbitrary integer powers for one type of symmetric anti-tridiagonal matrices of even order, Appl. Math. Comput. 204 (2008) 288–298. [3] J. Gutiérrez-Gutiérrez, Positive integer powers of certain tridiagonal matrices, Appl. Math. Comput. 202 (2008) 133–140. [4] R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1990.