3.2 Linear operators, matrices and polynomials

3.2.1 Linear operators and matrices

  • Definition. Let \(V\) be a vector space over \(\F \). A linear operator on \(V\) is a linear map \(\phi :V\to V\).

    The vector space of linear operators on \(V\) is denoted \(L(V)\) (instead of \(L(V,V)\)).

  • Notation. Write \(M_n(\F )\) for \(M_{n\times n}(\F )\).

  • Definition. Let \(V\) be a finite-dimensional vector space over \(\F \) with basis \(\cB :\lst {v}1n\). Let \(\phi \in L(V)\). The matrix of \(\phi \) with respect to \(\cB \) is the matrix \(A=(A_{ij})\in M_{n}(\F )\) defined by:

    \begin{equation} \label {eq:27} \phi (v_j)=\sum _{i=1}^nA_{ij}v_{i}, \end{equation}

    for all \(\bw 1jn\).

3.2.2 Polynomials in linear operators and matrices

  • Notation. For \(\phi ,\psi \in L(V)\) write \(\phi \psi \) for \(\phi \circ \psi \in L(V)\).

    Similarly, write \(\phi ^{n}\) for the \(n\)-fold composition of \(\phi \) with itself:

    \begin{equation*} \phi ^n=\phi \circ \dots \circ \phi , \end{equation*}

    with \(\phi \) repeated \(n\) times on the right, and define \(\phi ^0:=\id _V\), \(\phi ^1:=\phi \).

    Finally, for \(A\in M_n(\F )\), set \(A^0=I_{n}\), \(A^1=A\).

  • Definition. Let \(p\in \F [x]\), \(p=a_0+\dots +a_nx^n\), \(\phi \in L(V)\) and \(A\in M_n(\F )\). Then \(p(\phi )\in L(V)\) and \(p(A)\in M_n(\F )\) are given by:

    \begin{align*} p(\phi )&:= a_0\id _V+a_1\phi +\dots +a_n\phi ^n=\sum _{k\in \N }a_{k}\phi ^k,\\ p(A)&:= a_0I_{n}+a_1A+\dots +a_nA^n=\sum _{k\in \N }a_{k}A^k. \end{align*}

  • Proposition 3.4. For \(p,q\in \F [x]\), \(\phi \in L(V)\) and \(A\in M_n(\F )\),

    \begin{align} \label {eq:8} (p+q)(\phi )&=p(\phi )+q(\phi )&(p+q)(A)&=p(A)+q(A)\\ (pq)(\phi )&=p(\phi )q(\phi )=q(\phi )p(\phi )&(pq)(A)&=p(A)q(A)=q(A)p(A)\label {eq:9}. \end{align}