Diagonalize the Upper Triangular Matrix and Find the Power of the Matrix
Problem 583
Consider the $2\times 2$ complex matrix
\[A=\begin{bmatrix}
a & b-a\\
0& b
\end{bmatrix}.\]
(a) Find the eigenvalues of $A$.
(b) For each eigenvalue of $A$, determine the eigenvectors.
(c) Diagonalize the matrix $A$.
(d) Using the result of the diagonalization, compute and simplify $A^k$ for each positive integer $k$.
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