On invariant subspaces of matrices: a new proof of a theorem of Halmos (Q603155)

\textit{P. R. Halmos} [Linear Algebra Appl. 4, 11--15 (1971; Zbl 0264.15001)], proved that if \(A\) is an \(n\times n\) matrix over the field of complex numbers and \(E\) is an \(A\)-invariant subspace, then there exist matrices \(B\) and \(C\) such that \(BA=AB\), \(CA=AC\), \(E\) is the kernel of \(B\) and \(E\) is the range of \(C\). In this paper the author gives an elementary elegant proof of this result and shows that there exist \(B\) and \(C\) that additionally satisfy \(BC=CB=0\). The author remarks that this result also holds for a matrix \(A\) with coefficients in any field \(K\).