4.2 Invariant subspaces

  • Definition. Let \(\phi \) be a linear operator on a vector space \(V\). A subspace \(U\sub V\) is \(\phi \)-invariant if and only if \(\phi (u)\in U\), for all \(u\in U\).

  • Lemma 4.1. Let \(\phi ,\psi \in L(V)\) be linear operators and suppose that \(\phi \psi =\psi \phi \) (say that \(\phi \) and \(\psi \) commute).

    Then \(\ker \psi \) and \(\im \psi \) are \(\phi \)-invariant.

  • Definition. Let \(\lst {V}1k\leq V\) with \(V=\oplst {V}1k\) and let \(\phi _i\in L(V_i)\), for \(\bw 1ik\).

    Define \(\phi :V\to V\) by

    \begin{equation*} \phi (v)=\phi _1(v_1)+\dots +\phi _k(v_k), \end{equation*}

    where \(v=\plst {v}1k\) with \(v_i\in V_i\), for \(\bw 1ik\).

    Call \(\phi \) the direct sum of the \(\phi _i\) and write \(\phi =\oplst \phi 1k\).

  • Definition. Let \(\lst {A}1k\) be square matrices with \(A_i\in M_{n_i}(\F )\). The direct sum of the \(A_i\) is

    \begin{equation*} \oplst {A}1k:= \begin{pmatrix} A_1&&0\\&\ddots &\\0&&A_k \end {pmatrix}\in M_n(\F ), \end{equation*}

    where \(n=\plst {n}1k\).

    A matrix of this type is said to be block diagonal.

  • Proposition 4.2. Let \(\lst {V}1k\leq V\) with \(V=\oplst {V}1k\) and let \(\phi _i\in L(V_i)\), for \(\bw 1ik\). Let \(\phi =\oplst \phi 1k\). Then

    • (1) \(\phi \) is linear so that \(\phi \in L(V)\).

    • (2) Each \(V_i\) is \(\phi \)-invariant and \(\phi \restr {V_i}=\phi _i\), \(\bw 1ik\).

    • (3) Let \(\cB _i\) be a basis of \(V_i\) and \(\phi _i\) have matrix \(A_i\) with respect to \(\cB _i\), \(\bw 1ik\). Then \(\phi \) has matrix \(\oplst {A}1k\) with respect to the concatenated basis \(\cB =\cB _1\dots \cB _k\).

  • Proposition 4.3. Let \(\lst {V}1k\leq V\) with \(V=\oplst {V}1k\) and let \(\phi \in L(V)\). Suppose that each \(V_i\) is \(\phi \)-invariant.

    Then \(\phi =\oplst \phi 1k\) where \(\phi _i:=\phi \restr {V_i}\in L(V_i)\).

  • Proposition 4.4. Let \(\lst {V}1k\leq V\) with \(V=\oplst {V}1k\), \(\phi _i\in L(V_i)\), \(\bw 1ik\) and \(\phi =\oplst \phi 1k\).

    Then:

    • (1) \(\ker \phi =\oplst {\ker \phi }1k\).

    • (2) \(\im \phi =\oplst {\im \phi }1k\).

    • (3) \(p(\phi )=p(\phi _1)\oplus \dots \oplus p(\phi _k)\), for any \(p\in \F [x]\).

    • (4) \(\Delta _{\phi }=\prod _{i=1}^k\Delta _{\phi _i}\).

  • Proposition 4.5. Let \(\phi \in L(V)\) be a linear operator on a finite-dimensional vector space over a field \(\F \) and let \(\lst \lambda 1k\) be the distinct eigenvalues of \(\phi \).

    Then \(\phi \) is diagonalisable if and only if

    \begin{equation} \label {eq:21} V=\bigoplus _{i=1}^kE_{\phi }(\lambda _i). \end{equation}