00:01
Hello there.
00:01
Okay, so for this exercise we got this matrix, a, and we need to calculate the 10 power of this matrix.
00:09
So for that, we need to diagonalize this matrix.
00:12
That means that we can write a as p times d times p inverse, where d is a diagonal matrix, and b is an invertible matrix.
00:26
The point is that when you want to calculate a to the 10 power, using this diagonalization, it makes the things a lot easier because you only need to rise to the 10th power to the diagonal matrix.
00:41
And if the diagonal matrix is somehow like this, rising to the 10 power means just taking each element of the diagonal and rise them to the 10 power.
00:54
And that's all.
00:55
That's why this process makes a lot of, simplify a lot of things of calculations when you want to calculate the power of a matrix.
01:06
Of course this only works for diagonalizable matrix.
01:12
Okay, so let's do this.
01:14
So first we need to find that matrix d and p.
01:19
So let's start with the matrix d.
01:21
And the matrix d is a matrix that has the eigenvalue.
01:27
Of this matrix a on the diagonal.
01:31
In this case, it's a 2x2 matrix, so we only need to group them to eigenvalues.
01:38
So let's calculate the eigenvalues of this matrix.
01:44
So we need to get the determinant of this matrix, and this is equals to the characteristic polynomial, and sorry, i forgot.
01:54
This matrix, if you can observe, is lower triangular.
02:00
That means that the eigenvalues, we don't need to use this formula because the eigenvalues are located at the diagonal.
02:11
1 and 2.
02:12
So we got 1 and 2 are the corresponding eigenvalues.
02:18
That means that this diagonal matrix is going to be equals to 1, 2.
02:28
Great...