Exercise I.C.11

Question

Find $\phi,\varphi:\mathbb{R}^2	o\mathbb{R}^2$ such that both functions are nilpotent of order 2 and whose composition is idempotent.

Toğrul Topal · Accepted Answer

Set
    $$\phi \begin{bmatrix} x_1 \ x_2 \end{bmatrix} := \begin{bmatrix} x_1x_2 \ 0 \end{bmatrix} \qquad
    \varphi \begin{bmatrix} x_1 \ x_2 \end{bmatrix} := \begin{bmatrix} 0 \ x_1x_2 \end{bmatrix}$$
    Obviously, both functions are nilpotent of order 2. Furthermore, their composition is also a zero function, so applying it twice gives us zero function still, making $\phi \circ \varphi$ idempotent.

Exercise I.C.11

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Exercise I.C.11

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