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

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.

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=

 

=

 

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 =