Discussion
Consider an eigenvalue problem given by Ax=\(\lambda_{i}\)x. If \(\lambda_{i}\) represent the eigenvalues of the non-singular square matrix A, then what will be the eigenvalues of matrix A2?
Answer: B
The eigenvalues of \(A^{2}\) are \(\lambda_{i}^{2}\).
If \(Ax = \lambda_{i}x\), apply \(A\) again:
\[ \begin{aligned} A^{2}x &= A(\lambda_{i}x) \\ &= \lambda_{i}(Ax) = \lambda_{i}^{2}x \end{aligned} \]
So each eigenvector of \(A\) is an eigenvector of \(A^{2}\), with the eigenvalue squared. In general the eigenvalues of \(A^{k}\) are \(\lambda_{i}^{k}\). \(\lambda_{i}^{4}\) would belong to \(A^{4}\), and \(\lambda_{i}^{1/2}\) and \(\lambda_{i}^{1/4}\) to square and fourth roots of \(A\).
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