MA20216: Algebra 2A

4.4 Jordan normal form

4.4.1 Jordan blocks

Definition. The Jordan block of size \(n\in \Z _+\) and eigenvalue \(\lambda \in \F \) is \(J(\lambda ,n)\in M_n(\F )\) with \(\lambda \)’s on the diagonal, \(1\)’s on the super-diagonal and zeros elsewhere. Thus
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} J(\lambda ,n)= \begin{pmatrix} \lambda &1&0&\dots &0\\ &\ddots &\ddots &\ddots &\vdots \\ &&\ddots &\ddots &0\\ &&&\ddots &1\\ 0&&&&\lambda \end {pmatrix} \end{equation*}

Notation. Set \(J_n:=J(0,n)\) so that \(J(\lambda ,n)=\lambda I_n+J_n\).

Lemma 4.16. Let \(\lst {v}1n\) be a basis for a vector space \(V\) and \(\phi \in L(V)\).

Then the following are equivalent:
- (1) \(\phi \) has matrix \(J_n\) with respect to \(\lst {v}1n\).
- (2) \(\phi (v_1)=0\) and \(\phi (v_i)=v_{i-1}\), for \(\bw 2in\).
- (3) \(v_i=\phi ^{n-i}(v_n)\), \(\bw 0i{n-1}\) and \(\phi ^{n}(v_n)=0\).

Theorem 4.17. Let \(\phi \in L(V)\) be a nilpotent operator on a finite-dimensional vector space over \(\F \). Then there are \(\lst {v}1k\in V\) and \(\lst {n}1k\in \Z _+\) such that
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} \phi ^{n_1-1}(v_1),\dots ,\phi (v_1),v_1,\dots ,\phi ^{n_k-1}(v_k),\dots ,\phi (v_k),v_k \end{equation*}

is a basis of \(V\) and \(\phi ^{n_i}(v_i)=0\), for \(\bw 1ik\).

Corollary 4.18. Let \(\phi \in L(V)\) be a nilpotent operator on a finite-dimensional vector space over \(\F \). Then there is a basis for which \(\phi \) has matrix \(J_{n_1}\oplus \dots \oplus J_{n_k}\).

Proposition 4.19. Let \(\phi \in L(V)\) be nilpotent with matrix \(J_{n_1}\oplus \dots \oplus J_{n_k}\) for some basis of \(V\). Then \(\lst {n}1k\) are unique up to order. Indeed,
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} \#\set {i\st n_i\geq s}=\dim \ker \phi ^s-\dim \ker \phi ^{s-1}, \end{equation*}

for each \(s\geq 1\).

Proposition 4.20. In the situation of Proposition 4.19, we have
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} m_{\phi }=x^s, \end{equation*}

where \(s=\max \set {\lst {n}1k}\).

4.4.2 Jordan normal form

Theorem 4.21. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space \(V\) over \(\C \). Then there is a basis of \(V\) for which \(\phi \) has as matrix a direct sum of Jordan blocks which are unique up to order.

Such a basis is called a Jordan basis and the direct sum of Jordan blocks is called the Jordan normal form (JNF) of \(\phi \).

Corollary 4.22. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space \(V\) over \(\C \) with distinct eigenvalues \(\lst \lambda 1k\). Then
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} m_{\phi }=\prod _{i=1}^k(x-\lambda _i)^{s_i} \end{equation*}

where \(s_i\) is the size of the largest Jordan block of \(\phi \) with eigenvalue \(\lambda _{i}\).

Corollary 4.23. Any \(A\in M_n(\C )\) is similar to a direct sum of Jordan blocks, that is, there is an invertible matrix \(P\in M_n(\C )\) such that
\(\seteqnumber{0}{4.}{2}\)
\begin{equation*} P^{-1}AP=\oplst {A}1r, \end{equation*}

with each \(A_i\) a Jordan block.

\(\oplst {A}1r\) is called the Jordan normal form (JNF) of \(A\) and is unique up to the order of the \(A_i\).

Theorem 4.24. Matrices \(A,B\in M_n(\C )\) are similar if and only if they have the same Jordan normal form, up to reordering the Jordan blocks.