Matrices Problem

[Kwahha]

Hey,

I have a problem. I want to evaluate the following matrix problem:

1 2 2
2 1 2 <-- this is the matrix, and i want to multiply it by itself for n number of times (so: matrix^n)
2 2 3

How will the new matrix look with the entire matrix in terms of n?

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Kwahha

 

[Deleted member 4993]

Hey,

I have a problem. I want to evaluate the following matrix problem:

1 2 2
2 1 2 <-- this is the matrix, and i want to multiply it by itself for n number of times (so: matrix^n)
2 2 3

How will the new matrix look with the entire matrix in terms of n?

... ... ...
... ... ...
... ... ...

Kwahha

What do you get when multiply the matrix once by itself? What do you get when multiply the matrix twice by itself? see any useful pattern? Convince yourself by multiplying third time.

 

[Kwahha]

What do you get when multiply the matrix once by itself? What do you get when multiply the matrix twice by itself? see any useful pattern? Convince yourself by multiplying third time.

I did check those, however, i can not find a pattern in them.

n=1

1 2 2
2 1 2
2 2 3

n=2

9 8 12
8 9 12
12 12 17

n=3

49 50 70
50 49 70
70 70 99

n=4

289 288 408
288 289 408
408 408 577

n=5

1681 1682 2378
1682 1681 2378
2378 2378 3363

However, i didnt find sth for the upper left corner namely:
n=1 --> 1^2
n=2 --> 3^2
n=3 --> 7^2
n=4 --> 17^2
n=5 --> 41^2
n=6 --> 99^2
n=7 --> 239^2
n=8 --> 577^2;
But i dont know how to express those in terms of n

 

[HallsofIvy]

You could do this by finding the eigenvalues and corresponding eigenvectors for this matrix. That way, we could write \(\displaystyle A= PDP^{-1}\) where "A" is the given matrix, "D" is the diagonal matrix having the eigenvalues of A on its diagonal, and "P" is the matrix having the eigenvectors of A as columns. Then, \(\displaystyle A^n= (P^{-1}DP)^n= (P^{-1}DP)(P^{-1}DP)(P^{-1}DP)\cdot\cdot\cdot= P^{-1}D^nP\).

It looks like the eigenvalues are pretty easy to find (one is -1) but the eigenvectors might be harder.