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Minimal Polynomial Question

  1. Oct 22, 2009 #1
    1. The problem statement, all variables and given/known data

    Let V be the vector space of n x n matrices over the field F. Fix [tex] A \in V[/tex]. Let T be the linear operator on V defined by T(B) = AB, for all [tex] B \in V[/tex].

    a). Show that the minimal polynomial for T equals the minimal polynomial for A.
    b) Find the matrix of T with respect the the standard basis of V. i.e. the basis [tex]\left\{E_{ij} \right| 1 \leq i,j \leq n [/tex]}, where [tex]E_{ij}[/tex] is the matrix having 1 in the (i,j)th entry and zeros everywhere else.

    2. Relevant equations

    ...

    3. The attempt at a solution

    I know that the operator T is represented in some ordered basis by the matrix A, then T and A have the same minimal polynomial. The problem I'm running into is that I'm having a really hard time understanding abstract linear algebra, so this is all very very confusing to me and I'm not quite sure where to even start on this problem...
     
  2. jcsd
  3. Oct 23, 2009 #2

    HallsofIvy

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    What is the definition of "minimal polynomial" for a linear operator?
     
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