Let \(V\) be the vector space of \(n \times n\) matrices over the field \(F\). Let \(A\) be a fixed \(n \times n\) matrix. Let \(T\) be the linear operator on \(V\) defined by \[ T(B) = AB. \] Show that the minimal polynomial for \(T\) is the minimal polynomial for \(A\).
Solution
For every polynomial \(f(x)=\sum_{j=0}^r c_jx^j\) and every matrix \(B\), repeated application of \(T(B)=AB\) gives \[ T^j(B)=A^jB, \] and hence \[ f(T)(B)=\sum_{j=0}^r c_jA^jB=f(A)B. \] Therefore \(f(T)=0\) if \(f(A)=0\).
Conversely, if \(f(T)=0\), evaluate at \(B=I\): \[ 0=f(T)(I)=f(A)I=f(A). \] Thus the polynomials annihilating \(T\) are exactly the polynomials annihilating \(A\). Their monic generators are therefore equal, so \[ m_T=m_A. \]