For an integer \(r\geq 1\) and a diagonalizable matrix \(\Mtrx{A}\) with \(\Mtrx{D} = \Mtrx{C}^{-1} \Mtrx{A} \Mtrx{C}\) diagonal,
| \(\Mtrx{A}^r\) | \(=\) | \(\Mtrx{C} \Mtrx{D}^r \Mtrx{C}^{-1}\) |
The equality \(\Mtrx{D} = \Mtrx{C}^{-1} \Mtrx{A} \Mtrx{C}\) is equivalent to
| \(\Mtrx{A}\) | \(=\) | \(\Mtrx{C} \Mtrx{D} \Mtrx{C}^{-1}\) |
Therefore
| \(\Mtrx{A}^r\) | \(=\) | \(\left( \Mtrx{C} \Mtrx{D} \Mtrx{C}^{-1} \right)^r\) |
| \(\) | \(= \) | \(\left( \Mtrx{C} \Mtrx{D} \Mtrx{C}^{-1} \right) \left( \Mtrx{C} \Mtrx{D} \Mtrx{C}^{-1} \right)\ \cdots\ \left( \Mtrx{C} \Mtrx{D} \Mtrx{C}^{-1} \right)\) |
| \(\) | \(= \) | \(\Mtrx{C}\, \Mtrx{D}^r\, \Mtrx{C}^{-1}\) |
as claimed.