THE THREE-BAND MATRICES
*U]HJRU]%LHUQDW-XVW\QD%RU\Ğ,ORQD&DáXVLĔVND$JQLHV]ND6XUPD Institute of Mathematics, Czestochowa University of Technology, Poland
imi@imi.pcz.pl
1. Formulation of the problem
In practical problem with numeric description of heat flowing [1] (Fourier equa- tion), the diffusion (Fick equation), etc. there are large linear system equations with 3-band matrices. In the paper we give the direct models for the determinants (proper- ty 1 and 4) and the algebraic complement of that matrices (property 2 and 5). In the result we receive the direct models for solving the linear systems equations expressed by the 3-band matrices (property 3 and 6).
2. Solution of the problem Let
n n n
a b
b a b b a b
b a b
b a
A
» u
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»
¼ º
««
««
««
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¬ ª
0 0
0 0 0
0 0
0 0
0 0
0
(1)
be the 3-band matrix of dimension n.
Property 1. The determinant of the matrix A is given by the formulan
¸¸¹
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§
¸¸¹·
¨¨©§
¸¸¹
¨¨© ·
§ 2 2 4 4 6 6
3 3 2
2 1
det 1 n a b
b n a
b n a
a A
Wn n n n n n (2)
Inductive proof. W1 Supposea.
¸¸¹
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§
¸¸¹·
¨¨©§
¸¸¹
¨¨© ·
§ 2 2 4 4 6 6
3 3 2
2 1
det 1 k a b
b k a
b k a
a A
Wk k k k k k (3)
for k n1 and k n. Then
¸¸¹
¨¨© ·
§
¸¸¹·
¨¨©§
¸¸¹
¨¨© ·
§
1 1 2 3 4 3 4
1 2
1 3
2 2
1
1 n a b
b n a
b n a a W b aW
Wn n n n n n n (4)
This end the proof.
Property 2. The algebraic complement of the matrix A is given by the formulan
j n j l l l
j j
l j l j
j A bW W
A 1
12 (5)
where jt1 and 0dldn j. Proof. A simple observation.
Property 3. Let
n n n
n n
k
a b c
b a c
c a
b
c b
a b
c b
a
Z
u
p
0 0
0 0
0 0
0 0 0
1 3 2 1
(6)
We have
, )
1 ( )
1 (
0
2 1
1
1
1
¦
¦
n k
i
i k n i i k i k k k
i
i i k k i i k n
n W c b W W c bW
Z where W0 1 (7)
So, the linear system equations
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¼ º
««
««
««
««
¬ ª
»»
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¼ º
««
««
««
««
¬ ª
»»
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¼ º
««
««
««
««
¬ ª
n n n n
c c
c c c
x x
x x x
a b
b a b b a b
b a b
b a
1 3 2 1
1 3 2 1
0 0
0 0 0
0 0
0 0
0 0
0
(8)
gives
n k
i
k n
i
i k n i i k i k k i i k k i i k n
k W
W b c
W W b c
W
x
¦
1¦
1 0
2 1
1 ( 1)
) 1 (
(9)
for 1dkdn. Let
n n n
a b
b a b b a b
b a b
b a
B
» u
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»»
»»
»
¼ º
««
««
««
««
¬ ª
2 0 0
0 0 0
0 0
0 0
0 0
0
(10)
be the quasi 3-band matrix of dimension n.
Property 4. The determinant of the matrix Bn is given by the formula
¸¸¹
·
¨¨©
§
¸¸¹·
¨¨©§
¸¸¹
¨¨© ·
§
¸¸¹·
¨¨©§
...
2 4 3 1 1
3 2 1 0
2 2 2 4 4 6 6
b n a
b n a
b n a
n a
Vn n n n n (11)
Proof. We have
¸¸¹
¨¨© ·
§
¸¸¹·
¨¨©§
¸¸¹
¨¨© ·
§
2 2 2 4 4 6 6
2
2 4 3 1
3 2 0
2 n n a b
b n a
b n n a
n a W b W
Vn n n n n n n (12)
This end the proof.
Property 5. The algebraic complements of the matrix Bnare given by the formulas
1 n j l,
j l j l j l j
j B bW V
B where j 1,...,n1 and 0dldn j1 (13)
1,
j j n j n j
n A b W
B where j 1,...,n (14)
, 2
2 jn nj j1
n
j A b W
B where j 1,...,n1 (15)
Proof. A simple observation.
Property 6. Let
n n n
n n
k
a b c
b a c
c a b
c b a b
c b
a
U
u
p
2 0
0 0 0
0 0
0 0 0
1 3 2 1
(16)
i) For k 1,2,...,n1we have
, )
1 ( )
1 (
0
2 1
1
1
1
¦
¦
n k
i
i k n i i k i k k k
i
i i k k i i k n
n V c b W W c bV
U where V0 1(17)
ii) and for k n we have
1 1
1
1
1 3 2 1
) 1 ( 2
2 0 0
0 0 0 0
0 0 0
u
p
¦
n n ni
i i n n i i
n n n n
n k
W c W b c
c b
c a b
c b
a b
c b
a b
c b
a
(18)
So, the linear system equations
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¼ º
««
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¬ ª
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¼ º
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¬ ª
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¼ º
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¬ ª
n n n n
c c
c c c
x x
x x x
a b
b a b b a b
b a b
b a
1 3 2 1
1 3 2 1
2 0 0
0 0 0
0 0
0 0
0 0
0
(19)
gives
n k
i
k n
i
i k n i i k i k k i i k k i i k n
k V
V b c
W W b c
V
x
¦
1¦
1 0
2 1
1 ( 1)
) 1 (
(20) for k 1,2,...,n1and
n
n n n
i
i i n n i i
n V
W c W b c
x
1 1
1
) 1
1 (
2
¦
(21) for k n.
References
[1] Majchrzak E., 0HWRGDHOHPHQWyZEU]HJRZ\FKZSU]HSá\ZLHFLHSáD, Wydawnictwo Politechniki CzĊVWRFKRZVNLHM&]ĊVWRFKRZD
[2] Majchrzak E., Mochnacki B., Metody numeryczne. Podstawy teoretyczne, aspekty praktyczne i algorytmy:\GDZQLFWZR3ROLWHFKQLNLĝOąVNLHM*OLZLFH4.
[3] Mochnacki B., Suchy J.S., 0RGHORZDQLH L V\PXODFMD NU]HSQLĊFLD RGOHZyZ, PWN, Warszawa 1993.
[4] Mostowski A., Stark M., (OHPHQW\DOJHEU\Z\ĪV]HM, PWN, Warszawa 1970.
