Matrix derivation of the Fibonacci series (you can give up the matrix if you cannot understand it) and the Fibonacci Matrix

Source: Internet
Author: User

Matrix derivation of the Fibonacci series (you can give up the matrix if you cannot understand it) and the Fibonacci Matrix

I. Matrix Multiplication

Set the matrix A and B to meet the following requirements: the number of columns of A = the number of rows of B

Calculation rules for matrix multiplication:

Multiply each row of matrix A by each column of matrix B.

*

 

=

 

=

 

Ii. Matrix derivation of the Fibonacci series

 

First, we want

Fib [I] = Fib [I-1] + Fib [I-2];

So the I of the Fibonacci series is related to two numbers, Fib [I-1] + Fib [I-2 ].

 

Then we can set the first matrix.

M1 =

 

Because we need to use the Matrix to launch the nth entry of Fibonacci.

So we set the next item of M1 to M3.

Then M3 = (that is, let the subscript of M1 move one bit after the whole)

 

Now we need a Transitional Matrix M2 to implement this operation from M1 to M3.

Because M1 is a 1*2 matrix

M3 is also a 1*2 matrix

So M2 must be a 2*2 matrix.

(Cause: 1. the M1 column must be the same as the M2 row.

2. The number of M3 columns is 2)

Set M2 =

 

So

A * fi-2 + B * fi-1 = fi-1

C * fi-2 + d * fi-1 = fi ②

 

This statement does not seem to be available.

But let's think about it.

 

When a = 0 & B = 1

When c = 1 & d = 1 (Fib [I] = Fib [I-1] + Fib [I-2])

Therefore, we can easily obtain the M2 matrix.

M2 =

 

 

3. A simple example

Set fi = fi-1 + 2fi-2 + 3fi-3

Obtain M1, M2, and M3 according to the above method.

Tip: matrix derivation without Coefficients

 

 

 

 

 

 

 

 

 

(We recommend that you first check the problem)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution:

Set M1 =

 

M3 =

 

Then M2 must be a 3*3 matrix.

Set M2 =

Then a * fi-1 + B * fi-2 + c * fi-3 = fi = fi-1 + 2fi-2 + 3fi-3

D * fi-1 + e * fi-2 + f * fi-3 = fi-1

G * fi-1 + h * fi-2 + I * fi-3 = fi-2

 

Easy to get a = 1, B = 2, c = 3

D = 1, e = 0, f = 0

G = 0, h = 0, I = 0

So

M2 =

 

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