- Email: [email protected]
On computing of arbitrary positive integer powers for tridiagonal matrices with elements −1,0,0, … ,0 in principal and 1,1,1, … ,1 in neighbouring diagonals – I
Available online at www.sciencedirect.com
Applied Mathematics and Computation 193 (2007) 253–256 www.elsevier.com/locate/amc
On computing of arbitrary positive integer powers for tridiagonal matrices with elements 1,0,0, . . . ,0 in principal and 1,1,1, . . . ,1 in neighbouring diagonals – I Jonas Rimas Department of Applied Mathematics, Faculty of Fundamental Sciences, Kaunas University of Technology, Kaunas 51368, Lithuania
Abstract In this paper we derive the general expression of the lth power (l 2 N Þ for one type of tridiagonal matrices of arbitrary order. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Tridiagonal matrices; Eigenvalues; Eigenvectors; Jordan’s form; Chebyshev polynomials
1. Introduction Solving some difference, differential equation and delay differential equations we meet the necessity to compute the arbitrary positive integer powers of square matrices [1,2]. In this paper we derive the general expression of the lth power (l 2 N Þ for one type of tridiagonal matrices. Similar expressions for some types of symmetric circulant and tridiagonal matrices were derived in [3–6]. 2. Derivation of general expression Consider the nth order ðn 2 N ; n P 2Þ matrix B of the following type: 1 0 1 1 C B 1 0 1 0 C B C B C B 1 0 1 C: B¼B C B ... C B C B @ 0 1 0 1A 1
0
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.051
ð1Þ
254
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
The lth power ðl 2 N Þ of this matrix we will find using the expression Bl ¼ TJ l T 1 [7], where J is the Jordan’s form of B, T is the transforming matrix. Matrices J and T can be found provided eigenvalues and eigenvectors of the matrix B are known. The eigenvalues of B are defined by the characteristic equation jB kEj ¼ 0; here E is the identity Let us denote a 1 1 Dn ðaÞ ¼
ð2Þ matrix of the nth order. 1 a
1
1
a
1 ...
0
1
0 ; a 1 1 a
a 1 Dn ðaÞ ¼
1 a 1 1 a
0 1 ...
0
1 a 1
; 1 a
ð3Þ
here a 2 R. Then jB kEj ¼ Dn ðkÞ:
ð4Þ
From (3) follows: Dn ¼ ða 1ÞDn1 Dn2
ð5Þ
and Dn ¼ aDn1 Dn2 ðD2 ¼ a2 1; D1 ¼ a; D0 ¼ 1Þ; here Dn ¼ Dn ðaÞ;
Dn ¼ Dn ðaÞ. Solving difference equation (6), we obtain Dn ðaÞ ¼ U n a a Dn ðaÞ ¼ U n U n1 ; 2 2 here Un(x) is the nth degree Chebyshev polynomial of the second kind: U n ðxÞ ¼
sinðn þ 1Þ arccos x ; sin arccos x
1 6 x 6 1:
a 2
ð6Þ , ð7Þ
ð8Þ
All the roots of the polynomial Un(x) are included in the interval [1; 1] and can be found using the relation [8] xnk ¼ cos
kp ; nþ1
k ¼ 1; 2; 3; . . . ; n:
ð9Þ
Taking (4), (7) and (9) into account we find the roots of the characteristic equation (2) (the eigenvalues of the matrix B): ð2k 1Þp ; 2n þ 1
k ¼ 1; n: ð10Þ Since all the eigenvalues kk ; k ¼ 1; n are simple eigenvalues (lk ¼ 1; lk denotes the multiplicity of the eigenvalue kk) for each eigenvalue kk corresponds single Jordan cell J 1 ðkk Þ in the matrix J. Taking this into account we write down the Jordan’s form of the matrix B [7]: kk ¼ 2 cos
J ¼ diagðk1 ; k2 ; k3 ; . . . ; kn Þ:
ð11Þ
Using the equality J ¼ T 1 BT we find the matrices T, T1 and derive the expression of the lth power (l 2 N Þ of the matrix B: Bl ¼ TJ l T 1 ¼ QðlÞ ¼ ðqij ðlÞÞ;
ð12Þ
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
255
here qij ðlÞ ¼ tk ¼
tk klk U 2i3 2
k¼1 ( 2k n2kþ2 2nþ1 2k2kn1 2nþ1
m¼
n X
0; 1;
;
kk kk U 2j3 ; 2 2 2
i; j ¼ 1; n;
if k ¼ 1; nþm ; 2
ð13Þ
; if k ¼ nþ2þm ; n; 2 if n ¼ 2p ðp 2 N Þ; if n ¼ 2p þ 1 ðp 2 N Þ;
ð14Þ
kk k ¼ 1; n are the eigenvalues of the matrix B (defined by (10)), n is the order of the matrix B (n 2 N ; n P 2Þ; U k ðxÞ is the function, defined by (8). The matrix B of arbitrary order n (see (1)) is nonsingular since all its eigenvalues are nonzero: kk 6¼ 0; k ¼ 1; n; n 2 N ; n P 2 (see (10)). This means that derived expression (12) can be applied for computing negative integer powers of B. For example, taking l ¼ 1 in (12), we get following expression for the elements of the inverse matrix B1: n X tk kk kk 1 U 2i3 ð15Þ fB gij ¼ U 2j3 ; i; j ¼ 1; n; 2 2 k 2 2 k k¼1 here n 2 N ðn P 2Þ is the order of the matrix B; kk ; tk and Uk (x) are defined by (10), (13) and (8), respectively. 3. Numerical considerations We can find the arbitrary positive integer powers of the nth order matrix (1) ðn 2 N ; n P 2Þ taking into account derived expressions. For example, if n = 3, we would have: p a ¼ 2 cos ; 7
J ¼ diagðk1 ; k2 ; k3 Þ ¼ diagða; b; cÞ; 0 1 a1 a2 a 3 1 Bl ¼ ðqij Þ ¼ @ a2 a4 a5 A; 7 a3 a5 a 6
b ¼ 2 cos
3p ; 7
c ¼ 2 cos
ð16Þ
a1 ðlÞ ¼ ð2 þ aÞðaÞl þ ð2 þ bÞðbÞl þ ð2 cÞcl ; a2 ðlÞ ¼ bð2 þ aÞðaÞ
lþ1
cð2 þ bÞðbÞ
l
lþ1
þ að2 cÞclþ1 ;
l
a3 ðlÞ ¼ bð2 þ aÞðaÞ cð2 þ bÞðbÞ þ að2 cÞcl ; a4 ðlÞ ¼ b2 ð2 þ aÞðaÞ
lþ2
þ c2 ð2 þ bÞðbÞ
lþ2
þ a2 ð2 cÞclþ2 ;
a5 ðlÞ ¼ b2 ð2 þ aÞðaÞ
lþ1
þ c2 ð2 þ bÞðbÞ
lþ1
þ a2 ð2 cÞclþ1 ;
l
l
a6 ðlÞ ¼ b2 ð2 þ aÞðaÞ þ c2 ð2 þ bÞðbÞ þ a2 ð2 cÞcl : Assuming l ¼ 1 in (16) , 0 1 0 1 B 1 B ¼@ 0 0 1 1
we get the expression for the inverse matrix B1: 1 C A:
1 1
Taking l ¼ 2, we have 0
2
B 2 B2 ¼ ðB1 Þ ¼ @ 1 2
1 1 1
2
1
C 1 A: 3
2p ; 7
256
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
If n = 4, we would obtain: J ¼ diagðk1 ; k2 ; k3 ; k4 Þ ¼ diagða; 1; b; cÞ; p 4p 2p a ¼ 2 cos ; b ¼ 2 cos ; c ¼ 2 cos ; 9 9 9 0 1 a1 a2 a3 a4 B 1 B a2 a5 a6 a8 C C Bl ¼ ðqij Þ ¼ B C; 9 @ a3 a6 a8 a9 A a4
a7
a9
2
ð17Þ
a10 l
2
l
2
a1 ðlÞ ¼ ð2 cÞða þ 1Þ ðaÞ þ 3ð1Þ þ ð2 þ aÞð1 bÞ bl þ ð2 bÞðc 1Þ cl ; l
a2 ðlÞ ¼ ð2 cÞðc þ 1Þða þ 1ÞðaÞ þ ð2 þ aÞða 1Þð1 bÞbl þ ð2 bÞð1 þ bÞðc 1Þcl ; a3 ðlÞ ¼ ð2 cÞða þ 1ÞðaÞ
lþ1 l
l
3ð1Þ ð2 þ aÞð1 bÞblþ1 þ ð2 bÞðc 1Þclþ1 ; l
a4 ðlÞ ¼ ð2 cÞða þ 1ÞðaÞ þ 3ð1Þ ð2 þ aÞð1 bÞbl þ ð2 bÞðc 1Þcl ; 2
2
l
2
a5 ðlÞ ¼ ð2 cÞðc þ 1Þ ðaÞ þ ð2 þ aÞða 1Þ bl þ ð2 bÞð1 þ bÞ cl ; a6 ðlÞ ¼ ð2 cÞðc þ 1ÞðaÞ
lþ1
ð2 þ aÞða 1Þblþ1 þ ð2 bÞð1 þ bÞclþ1 ;
l
a7 ðlÞ ¼ ð2 cÞðc þ 1ÞðaÞ ð2 þ aÞða 1Þbl þ ð2 bÞð1 þ bÞcl ; a8 ðlÞ ¼ ð2 cÞðaÞlþ2 þ 3ð1Þlþ2 þ ð2 þ aÞblþ2 þ ð2 bÞclþ2 ; a9 ðlÞ ¼ ð2 cÞðaÞlþ1 þ 3ð1Þlþ1 þ ð2 þ aÞblþ1 þ ð2 bÞclþ1 ; a10 ðlÞ ¼ ð2 cÞðaÞl þ 3ð1Þl þ ð2 þ aÞbl þ ð2 bÞcl ; Assuming l ¼ 1 in (17) 0 0 1 0 B 1 1 0 B B1 ¼ B @ 0 0 0 1 1 1
, we get 1 1 1 C C C: 1 A 1
References [1] R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, 1992. [2] J. Rimas, Investigation of the dynamics of mutually synchronized systems, Telecommunications and Radio Engineer 32 (1977) 68–79. [3] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices – I, Applied Mathematics and Computation 165 (2005) 137–141. [4] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices – II, Applied Mathematics and Computation 169 (2005) 1016–1027. [5] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order – I, Applied Mathematics and Computation 168 (2005) 783–787. [6] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order – II, Applied Mathematics and Computation 172 (2006) 245–251. [7] R. Horn, Ch. Johnson, Matrix Analysis, Cambridge University Press, 1996. [8] L. Fox, J.B. Parke, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.
Applied Mathematics and Computation 193 (2007) 253–256 www.elsevier.com/locate/amc
On computing of arbitrary positive integer powers for tridiagonal matrices with elements 1,0,0, . . . ,0 in principal and 1,1,1, . . . ,1 in neighbouring diagonals – I Jonas Rimas Department of Applied Mathematics, Faculty of Fundamental Sciences, Kaunas University of Technology, Kaunas 51368, Lithuania
Abstract In this paper we derive the general expression of the lth power (l 2 N Þ for one type of tridiagonal matrices of arbitrary order. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Tridiagonal matrices; Eigenvalues; Eigenvectors; Jordan’s form; Chebyshev polynomials
1. Introduction Solving some difference, differential equation and delay differential equations we meet the necessity to compute the arbitrary positive integer powers of square matrices [1,2]. In this paper we derive the general expression of the lth power (l 2 N Þ for one type of tridiagonal matrices. Similar expressions for some types of symmetric circulant and tridiagonal matrices were derived in [3–6]. 2. Derivation of general expression Consider the nth order ðn 2 N ; n P 2Þ matrix B of the following type: 1 0 1 1 C B 1 0 1 0 C B C B C B 1 0 1 C: B¼B C B ... C B C B @ 0 1 0 1A 1
0
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.051
ð1Þ
254
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
The lth power ðl 2 N Þ of this matrix we will find using the expression Bl ¼ TJ l T 1 [7], where J is the Jordan’s form of B, T is the transforming matrix. Matrices J and T can be found provided eigenvalues and eigenvectors of the matrix B are known. The eigenvalues of B are defined by the characteristic equation jB kEj ¼ 0; here E is the identity Let us denote a 1 1 Dn ðaÞ ¼
ð2Þ matrix of the nth order. 1 a
1
1
a
1 ...
0
1
0 ; a 1 1 a
a 1 Dn ðaÞ ¼
1 a 1 1 a
0 1 ...
0
1 a 1
; 1 a
ð3Þ
here a 2 R. Then jB kEj ¼ Dn ðkÞ:
ð4Þ
From (3) follows: Dn ¼ ða 1ÞDn1 Dn2
ð5Þ
and Dn ¼ aDn1 Dn2 ðD2 ¼ a2 1; D1 ¼ a; D0 ¼ 1Þ; here Dn ¼ Dn ðaÞ;
Dn ¼ Dn ðaÞ. Solving difference equation (6), we obtain Dn ðaÞ ¼ U n a a Dn ðaÞ ¼ U n U n1 ; 2 2 here Un(x) is the nth degree Chebyshev polynomial of the second kind: U n ðxÞ ¼
sinðn þ 1Þ arccos x ; sin arccos x
1 6 x 6 1:
a 2
ð6Þ , ð7Þ
ð8Þ
All the roots of the polynomial Un(x) are included in the interval [1; 1] and can be found using the relation [8] xnk ¼ cos
kp ; nþ1
k ¼ 1; 2; 3; . . . ; n:
ð9Þ
Taking (4), (7) and (9) into account we find the roots of the characteristic equation (2) (the eigenvalues of the matrix B): ð2k 1Þp ; 2n þ 1
k ¼ 1; n: ð10Þ Since all the eigenvalues kk ; k ¼ 1; n are simple eigenvalues (lk ¼ 1; lk denotes the multiplicity of the eigenvalue kk) for each eigenvalue kk corresponds single Jordan cell J 1 ðkk Þ in the matrix J. Taking this into account we write down the Jordan’s form of the matrix B [7]: kk ¼ 2 cos
J ¼ diagðk1 ; k2 ; k3 ; . . . ; kn Þ:
ð11Þ
Using the equality J ¼ T 1 BT we find the matrices T, T1 and derive the expression of the lth power (l 2 N Þ of the matrix B: Bl ¼ TJ l T 1 ¼ QðlÞ ¼ ðqij ðlÞÞ;
ð12Þ
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
255
here qij ðlÞ ¼ tk ¼
tk klk U 2i3 2
k¼1 ( 2k n2kþ2 2nþ1 2k2kn1 2nþ1
m¼
n X
0; 1;
;
kk kk U 2j3 ; 2 2 2
i; j ¼ 1; n;
if k ¼ 1; nþm ; 2
ð13Þ
; if k ¼ nþ2þm ; n; 2 if n ¼ 2p ðp 2 N Þ; if n ¼ 2p þ 1 ðp 2 N Þ;
ð14Þ
kk k ¼ 1; n are the eigenvalues of the matrix B (defined by (10)), n is the order of the matrix B (n 2 N ; n P 2Þ; U k ðxÞ is the function, defined by (8). The matrix B of arbitrary order n (see (1)) is nonsingular since all its eigenvalues are nonzero: kk 6¼ 0; k ¼ 1; n; n 2 N ; n P 2 (see (10)). This means that derived expression (12) can be applied for computing negative integer powers of B. For example, taking l ¼ 1 in (12), we get following expression for the elements of the inverse matrix B1: n X tk kk kk 1 U 2i3 ð15Þ fB gij ¼ U 2j3 ; i; j ¼ 1; n; 2 2 k 2 2 k k¼1 here n 2 N ðn P 2Þ is the order of the matrix B; kk ; tk and Uk (x) are defined by (10), (13) and (8), respectively. 3. Numerical considerations We can find the arbitrary positive integer powers of the nth order matrix (1) ðn 2 N ; n P 2Þ taking into account derived expressions. For example, if n = 3, we would have: p a ¼ 2 cos ; 7
J ¼ diagðk1 ; k2 ; k3 Þ ¼ diagða; b; cÞ; 0 1 a1 a2 a 3 1 Bl ¼ ðqij Þ ¼ @ a2 a4 a5 A; 7 a3 a5 a 6
b ¼ 2 cos
3p ; 7
c ¼ 2 cos
ð16Þ
a1 ðlÞ ¼ ð2 þ aÞðaÞl þ ð2 þ bÞðbÞl þ ð2 cÞcl ; a2 ðlÞ ¼ bð2 þ aÞðaÞ
lþ1
cð2 þ bÞðbÞ
l
lþ1
þ að2 cÞclþ1 ;
l
a3 ðlÞ ¼ bð2 þ aÞðaÞ cð2 þ bÞðbÞ þ að2 cÞcl ; a4 ðlÞ ¼ b2 ð2 þ aÞðaÞ
lþ2
þ c2 ð2 þ bÞðbÞ
lþ2
þ a2 ð2 cÞclþ2 ;
a5 ðlÞ ¼ b2 ð2 þ aÞðaÞ
lþ1
þ c2 ð2 þ bÞðbÞ
lþ1
þ a2 ð2 cÞclþ1 ;
l
l
a6 ðlÞ ¼ b2 ð2 þ aÞðaÞ þ c2 ð2 þ bÞðbÞ þ a2 ð2 cÞcl : Assuming l ¼ 1 in (16) , 0 1 0 1 B 1 B ¼@ 0 0 1 1
we get the expression for the inverse matrix B1: 1 C A:
1 1
Taking l ¼ 2, we have 0
2
B 2 B2 ¼ ðB1 Þ ¼ @ 1 2
1 1 1
2
1
C 1 A: 3
2p ; 7
256
J. Rimas / Applied Mathematics and Computation 193 (2007) 253–256
If n = 4, we would obtain: J ¼ diagðk1 ; k2 ; k3 ; k4 Þ ¼ diagða; 1; b; cÞ; p 4p 2p a ¼ 2 cos ; b ¼ 2 cos ; c ¼ 2 cos ; 9 9 9 0 1 a1 a2 a3 a4 B 1 B a2 a5 a6 a8 C C Bl ¼ ðqij Þ ¼ B C; 9 @ a3 a6 a8 a9 A a4
a7
a9
2
ð17Þ
a10 l
2
l
2
a1 ðlÞ ¼ ð2 cÞða þ 1Þ ðaÞ þ 3ð1Þ þ ð2 þ aÞð1 bÞ bl þ ð2 bÞðc 1Þ cl ; l
a2 ðlÞ ¼ ð2 cÞðc þ 1Þða þ 1ÞðaÞ þ ð2 þ aÞða 1Þð1 bÞbl þ ð2 bÞð1 þ bÞðc 1Þcl ; a3 ðlÞ ¼ ð2 cÞða þ 1ÞðaÞ
lþ1 l
l
3ð1Þ ð2 þ aÞð1 bÞblþ1 þ ð2 bÞðc 1Þclþ1 ; l
a4 ðlÞ ¼ ð2 cÞða þ 1ÞðaÞ þ 3ð1Þ ð2 þ aÞð1 bÞbl þ ð2 bÞðc 1Þcl ; 2
2
l
2
a5 ðlÞ ¼ ð2 cÞðc þ 1Þ ðaÞ þ ð2 þ aÞða 1Þ bl þ ð2 bÞð1 þ bÞ cl ; a6 ðlÞ ¼ ð2 cÞðc þ 1ÞðaÞ
lþ1
ð2 þ aÞða 1Þblþ1 þ ð2 bÞð1 þ bÞclþ1 ;
l
a7 ðlÞ ¼ ð2 cÞðc þ 1ÞðaÞ ð2 þ aÞða 1Þbl þ ð2 bÞð1 þ bÞcl ; a8 ðlÞ ¼ ð2 cÞðaÞlþ2 þ 3ð1Þlþ2 þ ð2 þ aÞblþ2 þ ð2 bÞclþ2 ; a9 ðlÞ ¼ ð2 cÞðaÞlþ1 þ 3ð1Þlþ1 þ ð2 þ aÞblþ1 þ ð2 bÞclþ1 ; a10 ðlÞ ¼ ð2 cÞðaÞl þ 3ð1Þl þ ð2 þ aÞbl þ ð2 bÞcl ; Assuming l ¼ 1 in (17) 0 0 1 0 B 1 1 0 B B1 ¼ B @ 0 0 0 1 1 1
, we get 1 1 1 C C C: 1 A 1
References [1] R.P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, 1992. [2] J. Rimas, Investigation of the dynamics of mutually synchronized systems, Telecommunications and Radio Engineer 32 (1977) 68–79. [3] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices – I, Applied Mathematics and Computation 165 (2005) 137–141. [4] J. Rimas, On computing of arbitrary positive integer powers for one type of odd order symmetric circulant matrices – II, Applied Mathematics and Computation 169 (2005) 1016–1027. [5] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order – I, Applied Mathematics and Computation 168 (2005) 783–787. [6] J. Rimas, On computing of arbitrary positive integer powers for one type of symmetric tridiagonal matrices of even order – II, Applied Mathematics and Computation 172 (2006) 245–251. [7] R. Horn, Ch. Johnson, Matrix Analysis, Cambridge University Press, 1996. [8] L. Fox, J.B. Parke, Chebyshev Polynomials in Numerical Analysis, Oxford University Press, London, 1968.