Explicit inverses of some tridiagonal matrices - Estudo Geral ...
Explicit inverses of some tridiagonal matrices - Estudo Geral ...
Explicit inverses of some tridiagonal matrices - Estudo Geral ...
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14 C.M. da Fonseca, J. Petronilho / Linear Algebra and its Applications 325 (2001) 7–21<br />
i.e.,<br />
φ i = (−1) i+1 Q n+1−i (0),<br />
where<br />
Q 2i (x) = P i [π 2 (x)] , Q 2i+1 (x) = (x − a 2 )Pi ∗ [π 2 (x)] ,<br />
with π 2 (x), P i (x) and Pi ∗ (x) the same as in (11), (13) and (12), respectively. Observe<br />
that if a 1 = a 2 , then θ i = φ n+1−i .<br />
We have determined completely the θ i ’s and φ i ’s <strong>of</strong> (7 ) – thus the inverse <strong>of</strong> the<br />
matrix – in the case <strong>of</strong> a <strong>tridiagonal</strong> 2-Toeplitz matrix (1).<br />
Theorem 4.1. Let A be the <strong>tridiagonal</strong> matrix (1), with a 1 a 2 /= 0 and b 1 b 2 c 1 c 2 > 0.<br />
Put<br />
π 2 (x):=(x − a 1 )(x − a 2 ), β := √ b 2 c 2 /b 1 c 1<br />
and let {Q i (·; α, γ )} be the sequence <strong>of</strong> polynomials defined by<br />
(√ ) [ ( )<br />
i π2 (x) − b 1 c 1 − b 2 c 2<br />
Q 2i (x; α, γ )= b1 b 2 c 1 c 2 U i<br />
2 √ b 1 b 2 c 1 c<br />
( )] 2<br />
π2 (x) − b 1 c 1 − b 2 c 2<br />
+γU i−1<br />
2 √ b 1 b 2 c 1 c 2<br />
(√ ) ( )<br />
i π2 (x) − b 1 c 1 − b 2 c 2<br />
Q 2i+1 (x; α, γ ) = (x − α) b1 b 2 c 1 c 2 Ui<br />
2 √ ,<br />
b 1 b 2 c 1 c 2<br />
where α and γ are <strong>some</strong> parameters. Under these conditions,<br />
⎧<br />
⎪⎨ (−1) i+j b<br />
(A −1 p ⌊(j−i)/2⌋<br />
i<br />
) ij =<br />
⎪⎩<br />
(−1) i+j c ⌊(j−i)/2⌋<br />
p j<br />
b ⌊(j−i+1)/2⌋<br />
q i<br />
θ i−1 φ j+1 /θ n if i j,<br />
c ⌊(j−i+1)/2⌋<br />
q j<br />
θ j−1 φ i+1 /θ n if i>j,<br />
(14)<br />
where p l = (3 − (−1) l )/2, q l = (3 + (−1) l )/2, ⌊z⌋ denotes the greater integer<br />
less or equal to the real number z,<br />
θ i = (−1) i Q i (0; a 1 ,β) , (15)<br />
and<br />
⎧<br />
⎨(−1) i Q n+1−i (0; a 1 , 1/β)<br />
φ i =<br />
⎩<br />
(−1) i+1 Q n+1−i (0; a 2 ,β)<br />
if n is odd,<br />
if n is even.<br />
Remark. As we already noticed, under the conditions <strong>of</strong> Theorem 4.1, if n is odd<br />
and b 1 c 1 = b 2 c 2 , then θ i = φ n+1−i ;andifn is even and a 1 = a 2 , then also θ i =<br />
φ n+1−i .