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    Classification of square-zero $2 \times 2$ complex matrices

    E-HK-62-11

    Area
    Algebra
    Topics
    Characteristic Values, Linear Algebra
    Status
    draft

    Let \(N\) be a \(2 \times 2\) complex matrix such that \(N^2 = 0\). Prove that either \(N = 0\) or \(N\) is similar over \(C\) to \[ \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}. \]

    Solution

    Assume \(N\ne0\). Choose \(v\) with \(Nv\ne0\) and put \(w=Nv\). Then \(Nw=N^2v=0\). The vectors \(v,w\) are independent, since \(v=\lambda w\) would imply \(Nv=\lambda Nw=0\). In the ordered basis \((v,w)\), \[ Nv=w,\qquad Nw=0, \] so the matrix of \(N\) is \[ \begin{bmatrix}0&0\\1&0\end{bmatrix}. \]

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