Good day,
I am interest in math, and while doing the exercise. I can find the nth root of the matrix if it is diagonalizable, using the $A^n = P*Q*P^-$, where Q is the diagonal, and P are the eigenvectors. But I found that it is not possible if matrix is not diagonalizable. For example what if I have 2 same eigen vectors and A is 2x2 matrix. So it is not possible to solve it. I have found the matrix for this case: \begin{bmatrix} 8 & 1\\ -4 & 12 \end{bmatrix}
What if i want to find the 19th root of this matrix. I found the eigen values(10) and vectors(0.5 1). How to solve this kind of problems?