Main Index Number Theory Sequences Recurrent sequences Linear recurrent sequences Binary recurrent sequences Lucas’ sequences Fibonacci Numbers
Subject Index
comment on the page

Matrix Identities

The recursive definition of Fibonacci numbers can be immediately used to verify its following matrix form

(F(n + 2) F(n + 1)) = (1 1) (F(n + 1) F(n) ) ,      n >= 1. F(n + 1) F(n) 1 0 F(n) F(n - 1)(1)

Its repeated application to the left hand side gives

(F(n + 2) F(n + 1)) = (1 1)^(n + 1),      n >= 0. F(n + 1) F(n) 1 0(2)

If we write the identity  (2) in the form

(F(2 n + 1) F(2 n) ) = (1 1)^(2 n) = (1 1)^n (1 1)^n = (F(n + 1) F(n) ) (F(n + ... F(2 n - 1) 1 0 1 0 1 0 F(n) F(n - 1) F(n) F(n - 1)

the matrix multiplication gives immediately the identity

F(2 n - 1) = F^2(n) + F^2(n - 1),(3)

and the doubling one

F(2 n) = F(n + 1) F(n) + F(n) F(n - 1) = (F(n) + F(n - 1)) F(n) + F(n) F(n - 1) = F^2(n) + 2 F(n) F(n - 1) .(4)

The recursive definition yields also the following identity

(F(n + 1)/F(n)) = (1 1) (F(n)/F(n - 1)) 1 0(5)

which implies that

(F(n + 1)/F(n)) = (1 1)^n (1/0) ,      n >= 0. 1 0(6)

Another identity in the above spirit is

(0 1)^n = F(n - 1) (1 0) + F(n) (0 1) . 1 1 0 1 1 1

Cite this web-page as:

Štefan Porubský: Matrix Identities.

Page created  .