Theorem 5.7.1.
Suppose \(V\) is a complex vector space, and \(T:V\to V\) is a linear operator. Let \(\lambda_1,\ldots, \lambda_k\) denote the distinct eigenvalues of \(T\text{.}\) (We can assume \(V\) is real if we also assume that all eigenvalues of \(V\) are real.) Then:
-
Generalized eigenvectors corresponding to distinct eigenvalues are linearly independent.
-
\(\displaystyle V = G_{\lambda_1}(T)\oplus G_{\lambda_2}(T)\oplus \cdots \oplus G_{\lambda_k}(T)\)
-
Each restriction \((T-\lambda_j)|_{G_{\lambda_j}(T)}\) is nilpotent.