Let \(V\) be a finite-dimensional vector space over a field \(F\), and let \(\phi: V \to V\) be a linear transformation.
Prove that there exists a direct sum decomposition \[ V = U \oplus W, \] where \(U\) and \(W\) are \(\phi\)-invariant subspaces of \(V\), \(\phi|_U\) is nilpotent, and \(\phi|_W\) is invertible (nonsingular).
Solution
Goal: Prove Fitting’s Lemma for linear operators on finite-dimensional vector spaces using the stabilization of kernels and images of powers of \(\phi\).
1. Stabilization of kernel and image chains:
1.1. Let \(n = \dim_F V < \infty\).
1.2. Consider the ascending chain of kernels and the descending chain of images:\[\ker(\phi) \subseteq \ker(\phi^2) \subseteq \ker(\phi^3) \subseteq \cdots \subseteq V,\]\[V \supseteq \operatorname{Im}(\phi) \supseteq \operatorname{Im}(\phi^2) \supseteq \operatorname{Im}(\phi^3) \supseteq \cdots\]
1.3. Because \(\dim_F V = n < \infty\), the sequence of dimensions \(\dim_F \ker(\phi^m)\) is non-decreasing and bounded above by \(n\).
1.4. Thus the chain of kernels must stabilize: there exists an integer \(k \le n\) such that\[\ker(\phi^k) = \ker(\phi^{k+1}) = \ker(\phi^{k+2}) = \cdots = \ker(\phi^{2k}).\]
1.5. Define \(U = \ker(\phi^k)\) and \(W = \operatorname{Im}(\phi^k)\).
2. \(U\) and \(W\) are \(\phi\)-invariant subspaces:
2.1. For \(u \in U = \ker(\phi^k)\):\[\phi^k(\phi(u)) = \phi(\phi^k(u)) = \phi(0) = 0,\]so \(\phi(u) \in \ker(\phi^k) = U\).
2.2. For \(w \in W = \operatorname{Im}(\phi^k)\), write \(w = \phi^k(v)\) for some \(v \in V\):\[\phi(w) = \phi(\phi^k(v)) = \phi^k(\phi(v)) \in \operatorname{Im}(\phi^k) = W.\]
3. Direct sum decomposition \(V = U \oplus W\):
3.1. Intersection \(U \cap W = \{0\}\):- Let \(v \in U \cap W\).- Since \(v \in W = \operatorname{Im}(\phi^k)\), there exists \(x \in V\) such that \(v = \phi^k(x)\).- Since \(v \in U = \ker(\phi^k)\), \(\phi^k(v) = 0\), which gives \(\phi^{2k}(x) = 0\).- Thus \(x \in \ker(\phi^{2k})\).- By stabilization at index \(k\) (<1>1), \(\ker(\phi^{2k}) = \ker(\phi^k)\), so \(x \in \ker(\phi^k)\).- Therefore \(v = \phi^k(x) = 0\).- Hence \(U \cap W = \{0\}\).
3.2. Sum \(V = U + W\):- By the Rank-Nullity Theorem applied to the linear map \(\phi^k: V \to V\):\[\dim_F \ker(\phi^k) + \dim_F \operatorname{Im}(\phi^k) = \dim_F V.\]- Since \(U \cap W = \{0\}\), \(\dim_F(U \oplus W) = \dim_F U + \dim_F W = \dim_F V\).- A subspace of \(V\) with the same finite dimension as \(V\) must be \(V\) itself.- Thus \(V = U \oplus W\).
4. Properties of the restrictions \(\phi|_U\) and \(\phi|_W\):
4.1. \(\phi|_U\) is nilpotent:- For every \(u \in U = \ker(\phi^k)\), \((\phi|_U)^k(u) = \phi^k(u) = 0\).- Thus \((\phi|_U)^k = 0\), so \(\phi|_U\) is nilpotent.
4.2. \(\phi|_W\) is invertible:- Since \(W\) is finite-dimensional, \(\phi|_W: W \to W\) is invertible if and only if \(\ker(\phi|_W) = \{0\}\).- Let \(w \in \ker(\phi|_W)\). Then \(w \in W\) and \(\phi(w) = 0\).- Then \(\phi^k(w) = \phi^{k-1}(\phi(w)) = \phi^{k-1}(0) = 0\), so \(w \in \ker(\phi^k) = U\).- Thus \(w \in U \cap W\).- By 3, \(U \cap W = \{0\}\), so \(w = 0\).- Therefore \(\ker(\phi|_W) = \{0\}\), which proves that \(\phi|_W\) is invertible.
5. Conclusion:
$V = U \oplus W$ with $U = \ker(\phi^k)$ and $W = \operatorname{Im}(\phi^k)$ is the required Fitting decomposition.