MA20216: Algebra 2A

4.3 Jordan decomposition

4.3.1 Powers of operators and Fitting’s Lemma

Proposition 4.6 (Increasing kernels, decreasing images). Let \(V\) be a vector space over a field \(\F \) and \(\phi \in L(V)\). Then
- (1) \(\ker \phi ^k\leq \ker \phi ^{k+1}\), for all \(k\in \N \). That is,
  \(\seteqnumber{0}{4.}{1}\)
  \begin{equation*} \set {0}=\ker \phi ^0\leq \ker \phi \leq \ker \phi ^2\leq \dots . \end{equation*}
  
  If \(\ker \phi ^k=\ker \phi ^{k+1}\) then \(\ker \phi ^k=\ker \phi ^{k+n}\), for all \(n\in \N \).
- (2) \(\im \phi ^k\geq \im \phi ^{k+1}\), for all \(k\in \N \). That is,
  \(\seteqnumber{0}{4.}{1}\)
  \begin{equation*} V=\im \phi ^0\geq \im \phi \geq \im \phi ^2\geq \dots . \end{equation*}
  
  If \(\im \phi ^k=\im \phi ^{k+1}\) then \(\im \phi ^k=\im \phi ^{k+n}\), for all \(n\in \N \).

Corollary 4.7. Let \(V\) be finite-dimensional with \(\dim V=n\) and \(\phi \in L(V)\). Then, for all \(k\in \N \),
\(\seteqnumber{0}{4.}{1}\)
\begin{align*} \ker \phi ^n&=\ker \phi ^{n+k}\\ \im \phi ^n&=\im \phi ^{n+k}. \end{align*}

Theorem 4.8 (Fitting¹’s Lemma). Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space over a field \(\F \). Then, with \(n=\dim V\), we have
\(\seteqnumber{0}{4.}{1}\)
\begin{equation*} V=\ker \phi ^n\oplus \im \phi ^n. \end{equation*}

¹ Hans Fitting, 1906–1938.

4.3.2 Generalised eigenspaces

Definition. Let \(\phi \in L(V)\) be a linear operator on a vector space over a field \(\F \). A generalised eigenvector of \(\phi \) with eigenvalue \(\lambda \) is \(v\in V\) such that, for some \(k\in \N \),
\(\seteqnumber{0}{4.}{1}\)
\begin{equation} \label {eq:6} (\phi -\lambda \id )^k(v)=0. \end{equation}

The set of all such is called the generalised eigenspace of \(\phi \) with eigenvalue \(\lambda \) and denoted \(G_{\phi }(\lambda )\). Thus
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} G_{\phi }(\lambda )=\set {v\in V\st (\phi -\lambda \id )^k(v)=0, \text {for some $k\in \N $}}= \bigcup _{k\in \N }\ker (\phi -\lambda \id _{V})^{k}. \end{equation*}

Lemma 4.9. \(E_{\phi }(\lambda )\leq G_{\phi }(\lambda )\leq V\) and \(G_{\phi }(\lambda )\) is \(\phi \)-invariant.

Lemma 4.10. Let \(\phi \in L(V)\) be a linear operator on a vector space over \(\F \) and \(\lambda _1,\lambda _2\in \F \) distinct eigenvalues of \(\phi \).

Then \(G_{\phi }(\lambda _1)\cap G_{\phi }(\lambda _2)=\set 0\).

Theorem 4.11 (Jordan² decomposition). Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space over \(\C \) with distinct eigenvalues \(\lst \lambda 1k\). Then
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} V=\bigoplus _{i=1}^kG_{\phi }(\lambda _i). \end{equation*}

² Camille Jordan, 1838–1922.

Definition. A linear operator \(\phi \) on a vector space \(V\) is nilpotent if \(\phi ^k=0\), for some \(k\in \N \). or, equivalently, if \(\ker \phi ^k=V\).

Proposition 4.12. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space \(V\) over \(\F \).

Then \(\phi \) is nilpotent if and only if there is a basis with respect to which \(\phi \) has a strictly upper triangular matrix \(A\) (thus \(A_{ij}=0\) whenever \(i\geq j\)):
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} A= \begin{pmatrix} 0&&*\\&\ddots &\\0&&0 \end {pmatrix}. \end{equation*}

Proposition 4.13. Let \(\lambda \in \C \) be an eigenvalue of a linear operator \(\phi \) on a complex finite-dimensional vector space. Then
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} \am (\lambda )=\dim G_{\phi }(\lambda ). \end{equation*}

Proposition 4.14. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space over \(\C \) with distinct eigenvalues \(\lst \lambda 1k\). Set \(\phi _i=\phi \restr {G_{\phi }(\lambda _i)}\). Then
- (1) Each \(m_{\phi _i}=(x-\lambda _i)^{s_i}\), for some \(s_i\leq \dim G_{\phi }(\lambda _i)\).
- (2) \(m_{\phi }=\prod _{i=1}^km_{\phi _i}=\prod _{i=1}^k(x-\lambda _i)^{s_i}\).

Corollary 4.15. Let \(\phi \in L(V)\) be a linear operator with minimum polynomial \(\prod _{i=1}^k(x-\lambda _i)^{s_i}\). Then
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} G_{\phi }(\lambda _i)=\ker (\phi -\lambda _i\id _V)^{s_i}. \end{equation*}