Mathematics Subject Classification (2020). 15A23, 11B39, 15A18,15A42
Many recurrence sequences of integers play a vital role in science and engineering. The zeros of characteristic polynomial of Fibonacci sequence matrix is always a fixed number
D. Fathima, M. m Albaidani, A. H. Ganie and A. Akhter introduced about New structure of Fibonacci numbers using concept of
In this paper, In Section 2, we recall the definition of Jacobsthal numbers and we investigate some properties of these matrix formed by Jacobsthal numbers. In Section 3, we define Jacobsthal-Lucas numbers and we investigate some properties of these matrices formed by these numbers. In section 4, we investigate the upper and lower bounds of the eigenvalues of these matrix by using the concept of majorization techniques.
For
We can extend Jacobsthal sequences through negative values of
From (1) and (2), we have listed the few values of Jacobsthal number in the table
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
… |
|
0 |
1 |
1 |
3 |
5 |
11 |
21 |
43 |
85 |
171 |
341 |
… |
|
0 |
|
|
|
|
|
|
|
|
|
|
… |
Based on Jacobsthal number we define Jacobsthal matrix
where
Consider the identity matrix 𝐼𝑛 of order 𝑛×𝑛. We define the matrices
and
Now we define a factor matrix as
By using the matrices
Generalized norm of Jacobsthal vector is defined by
The Jacobsthal matrix
For example,
The other factorization of
We define
For
Let
holds.
Proof. Let
Then
For example,
The inverses of the matrices
We know that
Define
Then
and
Now inverses of the Jacobsthal matrix is given by
Here we find the inverses of the Jacobsthal matrix by using the matrices
The inverses of the Jacobsthal matrix
For example,
For
We can extend Jacobsthal sequences through negative values of
From (1) and (2), we have listed the few values of Jacobsthal-Lucas number in the Table 2
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
… |
|
2 |
1 |
5 |
7 |
17 |
31 |
65 |
127 |
257 |
511 |
1025 |
… |
|
-2 |
|
|
|
|
|
|
|
|
|
|
… |
Based on Jacobsthal-Lucas number we define Jacobsthal-Lucas matrix
where
Consider the identity matrix
and
Now we define a factor matrix as
By using the matrices
The Jacobsthal matrix
For example,
The other factorization of
We define
For
Let
holds.
Proof. Let
Then
For example,
The inverses of the matrices
We know that
Define
Then
and
Now inverses of the Jacobsthal-Lucas matrix is given by
For
by
Proof. Together with the facts
This gives the Cholesky factorization of the matrix
For example, the Cholesky factorization of the inverses symmetric Jacobsthal matrix is given by
In this section we study about the eigenvalues of the symmetric Jacobsthal-Lucas matrix
Let
Let
When
Hardy, J E Littlewood, and G Polya introduced the concept of majorization techniques.
Let
Clearly
Let
Proof. Let
Since
Hence the result is completed.
Let
Then we have
For
Proof. Let
where
and
Then for
and
Hence
And also in addition, we have
and
The following lemma gives the lower bounds of the eigenvalues of
Jacobsthal-Lucas symmetric matrix
For
Proof. From
Clearly, the sum of the eigenvalues of the inverse symmetric Jacobsthal-Lucas matrix is positive and we get
For
In particular,
Proof. From Theorem 3.3.4 we have
we have
Then we get
By lemma 3.3.5, we have
Then we have
We present the decomposition of Jacobsthal matrix and Jacobsthal-Lucas symmetric matrix and their inverses using factor matrix. These matrices support to study about the medical image processing and cryptography for coding and decoding messages. Also we studied about the upper and lower of the eigen values of Jacobsthal-Lucas symmetric matrix.