Application of diagonalization for computing higher powers of a matrix

For diagonalizing a matrix A, the first step is to find the eigen values of it. Then find the corresponding eigen vectors of it. All the eigen vectors should be linearly independent if you want to diagonalize a matrix A. Otherwise, A is not diagonalizable. If you get the distinct eigen values then by the theorem eigen vectors corresponding to distinct eigen values are linearly independent. But if the eigen values are repeated, then the corresponding eigen vectors can be or cant be linearly independent. The diagonal matrix we obtain after the process contains the eigen values in the main diagonal.
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