Netted Binomial Matrices

In this paper, we define the bi-periodic Fibonacci matrix sequence that represent bi-periodic Fibonacci numbers. Then, we investigate generating function, Binet formula and summations of bi-periodic Fibonacci matrix sequence. After that, we say that some behaviours of bi-periodic Fibonacci numbers also can be obtained by considering properties of this new matrix sequence. Finally, we express that well-known matrix sequences, such as Fibonacci, Pell, $k$-Fibonacci matrix seque...

This is a survey on certain results which bring about a connection between Fibonacci sequences on the one hand and the areas of matrix theory and quantum information theory, on the other.

In this paper, we consider the matrix polynomial obtained by using bi-periodic Fibonacci matrix polynomial. Then, we give some properties and binomial transforms of the new matrix polynomials.

In this paper, we obtain a general expression for the entries of the r. (r is integer) power of a certain n-square complex tridiagonal matrix. In addition, we get the complex factorizations of Fibonacci polynomials, Fibonacci and Pell numbers.

We explore a certain family $\{A_n\}_{n=1}^{\infty}$ of $n \times n$ tridiagonal real symmetric matrices. After deriving a three-term recurrence relation for the characteristic polynomials of this family, we find a closed form solution. The coefficients of these characteristic polynomials turn out to involve the diagonal entries of Pascal's triangle in a tantalizingly predictive manner. Lastly, we explore a relation between the eigenvalues of various members of the family. Mo...

In this paper we study some further properties of the matrix with entries binom{i-1}{n-j}. We find the generating function for each row and column, and we find the eigenvalues and eigenvectors of this matrix. We also find the spectral properties of this matrix modulo 3 and 5.

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we investigate relationships between one type of graph and well-known Fibonacci sequence. In this content, we consider the adjacency matrix of one type of graph with 2k (k=1,2,...) vertices. It is also known that for any positive integer r, the (i,j)th entry of A^{r} (A is the adjacency matrix of the graph) is just the number of walks from vertex i to ve...

In this paper we use well-known results from linear algebra as tools to explore some properties of products of Fibonacci numbers. Specifically, we explore the behavior of the eigenvalues, eigenvectors, characteristic polynomials, determinants, and the norm of non-symmetric matrices embedded in the Hosoya triangle. We discovered that most of these objects either embed again in the Hosoya triangle or they give rise to Fibonacci identities. We also study the nature of these ma...

In this paper, we give some determinantal and permanental representations of Generalized Fibonacci Polynomials by using various Hessenberg matrices. These results are general form of determinantal and permanental representations of k sequences of the generalized order-k Fibonacci and Pell numbers.

In this paper, we consider the new family of recurrence sequences of $(q,k)$-generalized Fibonacci numbers. These sequences naturally extend the well-known sequences of $k$-generalized Fibonacci numbers and generalized $k$-order Pell numbers. We shall obtain a Binet-style formula and study the asymptotic behavior of dominant root of characteristic equation. Moreover, we shall prove some auxiliary results about these sequences. In particular, we characterize the first $(q,k)$-...