Some phenomenon of the powers of certain tridiagonal and asymmetric matrices

doi:10.1016/j.amc.2008.02.023

Applied Mathematics and Computation

Volume 202, Issue 1, 1 August 2008, Pages 348-359

https://doi.org/10.1016/j.amc.2008.02.023 Get rights and content

Abstract

In this paper, a new method of calculation of real (and even complex) powers of some asymmetric matrices obeying constance tridiagonals is presented.

Introduction

Let us set $T_{n} (a, b, c) ≔ {[\begin{matrix} a & c \\ b & a & c & 0 \\ b & a & c \\ ⋱ & ⋱ & ⋱ \\ 0 & b & a & c \\ b & a \end{matrix}]}_{n \times n},$ where $a, b, c \in C$ , $n \in N$ .

In the course of preparing paper [1] focused on a certain property of tridiagonal matrices and general constant-diagonals matrices, the authors came across a certain phenomenon connected with the real powers of tridiagonal matrix $T_{n} (x, 1, 1)$ and then with the complex powers of matrices of a more general form: $F_{3} (a, b, c, A) ≔ [\begin{matrix} a & b & c \\ b & A & b \\ c & b & a \end{matrix}],$ where $a, b, c \in C$ and $A ≔ a + c$ . This phenomenon consists of finding Binet’s formulas of elements of positive integers powers of given matrix and next to extend these ones to complex powers. Our considerations are extended also to matrices of asymmetric forms: $G_{3} (a, b, c, A, y) ≔ [\begin{matrix} a & yb & y^{2} c \\ b & A & yb \\ c & b & a \end{matrix}]$ and $G_{4} (a, b, c, d, e, A, y) ≔ [\begin{matrix} a & yb & y^{2} c & y^{3} e \\ b & A & yd & y^{2} c \\ c & d & A & yb \\ e & c & b & a \end{matrix}]$ for any $a, b, c, e, y \in C$ , $A ≔ a + yc$ , $d ≔ b + ye$ .

Section snippets

Positive integers powers of $F_{3}$

We shall start our deliberations with a simple case, that is with describing the positive integers powers of matrix (2).

Lemma 2.1

Let $a_{1}, b_{1}, c_{1} \in C$ and $A_{1} ≔ a_{1} + c_{1}$ . Then we have: ${(F_{3} (a_{1}, b_{1}, c_{1}, A_{1}))}^{k} = F_{3} (a_{k}, b_{k}, c_{k}, A_{k})$ for every $k \in N$ , where the following formulas hold: $A_{k} = a_{k} + c_{k},$ $a_{k + 1} = a_{1} a_{k} + b_{1} b_{k} + c_{1} c_{k},$ $b_{k + 1} = A_{1} b_{k} + A_{k} b_{1},$ $c_{k + 1} = a_{1} c_{k} + b_{1} b_{k} + c_{1} a_{k},$ $A_{k + 1} = 2 b_{1} b_{k} + A_{1} A_{k} .$

The proof of Lemma by induction follows.

Corollary 2.2

We have $a_{k} = c_{k} + (a_{1} - c_{1})^{k} .$

Proof

We note that (7), (8), (9) ⇒ a_k+1 − c_k+1 = (a₁ − c₁)(a_k − c_k), k = 1,2,… □

Corollary 2.3

Now let us set $(x, y \in C)$ : $z ≔ a_{1} - c_{1}, A_{1} ≔ \frac{1}{2} (x + y), b_{1} ≔ \frac{\sqrt{2}}{4} (x - y) .$

Phenomenon of the identity (5)

The identity (5) also holds for all $k \in C!$ Why is that so? First it should be noted that if in formulas (12), (13), (14), (15) we assume that $x ≔ x_{0}^{m / n}, y ≔ y_{0}^{m / n}, z ≔ z_{0}^{m / n},$ $m \in Z$ , $n \in 2 N - 1 ≔ {1, 3, 5, \dots}$ , $x_{0}, y_{0}, z_{0} \in R$ , then, by view of (5) we obtain $(F_{3} (a_{1} (x, y, z), b_{1} (x, y), c_{1} (x, y, z), A_{1} (x, y)))^{nr} = F_{3} (a_{mr} (x_{0}, y_{0}, z_{0}), b_{mr} (x_{0}, y_{0}), c_{mr} (x_{0}, y_{0}, z_{0}), A_{mr} (x_{0}, y_{0})),$ which means that $(F_{3} (a_{1} (x, y, z), b_{1} (x, y), c_{1} (x, y, z), A_{1} (x, y))) \in \sqrt[nr]{F_{3} (a_{mr} (x_{0}, y_{0}, z_{0}), b_{mr} (x_{0}, y_{0}), c_{mr} (x_{0}, y_{0}, z_{0}), A_{mr} (x_{0}, y_{0}))}$ for every $r \in N$ .

Now, again from the formulas (5) and (12), (13)

An asymmetric generalization of $T_{3} (x, 1, 1)$

We shall now designate the form of the powers of the matrix $T_{3} (x, 1, y)$ . By easily induction argument it can be deduced that the following identities hold: ${(T_{3} (x, 1, y))}^{k} = [\begin{matrix} a_{k}^{'} & {yb}_{k}^{'} & y (A_{k}^{'} - a_{k}^{'}) \\ b_{k}^{'} & A_{k}^{'} & {yb}_{k}^{'} \\ c_{k}^{'} & b_{k}^{'} & a_{k}^{'} \end{matrix}]$ for every $k \in N$ , where $a_{1}^{'} = A_{1}^{'} = x, b_{1}^{'} = 1, c_{1}^{'} = 0$ and $a_{k + 1}^{'} = {xa}_{k}^{'} + {yb}_{k}^{'},$ $b_{k + 1}^{'} = {xb}_{k}^{'} + A_{k}^{'},$ $c_{k + 1}^{'} = {xc}_{k}^{'} + b_{k}^{'},$ $A_{k + 1}^{'} = {xA}_{k}^{'} + 2 {yb}_{k}^{'},$ for every $k \in N$ . We note that also $A_{k}^{'} = a_{k}^{'} + {yc}_{k}^{'}, k \in N .$ From (29) we have $b_{k}^{'} = c_{k + 1}^{'} - {xc}_{k}^{'},$ which by (28) implies $A_{k}^{'} = b_{k + 1}^{'} - {xb}_{k}^{'} = c_{k + 2}^{'} - 2 {xc}_{k + 1}^{'} + x^{2} c_{k}^{'} .$ At last, from (30), (32), (33) we get $c_{k + 3}^{'} - 2 {xc}_{k + 2}^{'} + x^{2} c_{k +}$

Positive integers powers of $G_{4}$

In this section, we generalize the formulas (39), (40), (41), (42), (43), (44), (45), (46), (47), (48) for $G_{3}$ to the respective formulas for the matrix $G_{4}$ .

Lemma 5.1

Let $a_{1}, b_{1}, c_{1}, e_{1}, y, \in C$ and $A_{1} ≔ a_{1} + {yc}_{1} and d_{1} ≔ b_{1} + {ye}_{1} .$ Then we get ${(G_{4} (a_{1}, b_{1}, c_{1}, d_{1}, e_{1}, A_{1}, y))}^{k} = G_{4} (a_{k}, b_{k}, c_{k}, d_{k}, e_{k}, A_{k}, y)$ for every $k \in N$ , where the following recurrence formulas hold (again by induction argument): $A_{k} = a_{k} + {yc}_{k},$ $d_{k} = b_{k} + {ye}_{k},$ $a_{k + 1} = a_{1} a_{k} + {yb}_{1} b_{k} + y^{2} c_{1} c_{k} + y^{3} e_{1} e_{k},$ $b_{k + 1} = b_{1} a_{k} + A_{1} b_{k} + {yd}_{1} c_{k} + y^{2} c_{1} e_{k},$ $c_{k + 1} = c_{1} a_{k} + d_{1} b_{k} + A_{1} c_{k} + {yb}_{1} e_{k},$ $e_{k + 1} = e_{1} a_{k} + c_{1} b_{k} + b_{1} c_{k} + a_{1} e_{k} .$

Our next goal is to

Final remarks

1.
Recall that the definitions of complex powers of matrices $F_{3}$ , $G_{3}$ and $G_{4}$ presented in this paper are consequences of extensions of Binet’s formulas for elements of positive powers of matrices $F_{3}$ , $G_{3}$ and $G_{4}$ . It is the alternative and more effective way than the Jordan decomposition method (see [5]).
2.
The cases of matrices of the dimensions more than or equal to five will be discussed in the separated papers.
3.
A nice complement to the problem of diagonalization of a given matrix would be a paper [6].

References (6)

R. Wituła et al.
On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices
Appl. Math. Comput.
(2007)
W.G. Kelley et al.
Difference Equations. An Introduction with Applications
(2001)
C. Levesque
A basis of the set of sequences satisfying a given mth order linear recurrence
Elem. Math.
(1987)

There are more references available in the full text version of this article.

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Some phenomenon of the powers of certain tridiagonal and asymmetric matrices

Abstract

Introduction

Section snippets

Positive integers powers of F3

Phenomenon of the identity (5)

An asymmetric generalization of T3(x,1,1)

Positive integers powers of G4

Final remarks

Appl. Math. Comput.

Difference Equations. An Introduction with Applications

A basis of the set of sequences satisfying a given mth order linear recurrence

Elem. Math.

Positive integers powers of $F_{3}$

An asymmetric generalization of $T_{3} (x, 1, 1)$

Positive integers powers of $G_{4}$