1. Diagonalization

Let 0-2 46 .
q: Find formulas for the entries of , where is a positive integer.

I found eigenvalues, 2 and 4, then found their corresponding eigenspaces.

D^n=(P^-1)(A^n)P
=
2^n-2 0
8^n
.

2. Originally Posted by krtica
Let 0-2 46 .
q: Find formulas for the entries of , where is a positive integer.

I found eigenvalues, 2 and 4, then found their corresponding eigenspaces.

D^n=(P^-1)(A^n)P
=
2^n-2 0
8^n
.
What does this notation mean?

3. I apologize, the format didn't turn out as I expected. It is a 2x2 matrix. The first row is 2^n, -2. The second row is 0, 8^n.

D=(P^-1)AP, where P is a basis for both Eigenspaces with eigenvalues of 2 and 4. A is for the original matrix.

4. $\displaystyle M=\begin{bmatrix} 0 & -2\\ 4 & 6 \end{bmatrix}$ and $\displaystyle D^n=\begin{bmatrix} 2^n & -2\\ 0 & 8^n \end{bmatrix}$ correct?

5. Yes, that's correct.

6. $\displaystyle M^n=\begin{bmatrix} -1 & -1\\ 2 & 1 \end{bmatrix}\begin{bmatrix} 4 & 0\\ 0 & 2 \end{bmatrix}^n\begin{bmatrix} -1 & -1\\ 2 & 1 \end{bmatrix}^{-1}$

7. $\displaystyle M^n=XD^nX^{-1}$ where D is the a diagonal matrix of the eigenvalues and X is the matrix with the corresponding eigenvectors.

8. Thank you. I really, really appreciate your help.