# Is there any idempotent matrix that is not normal?

My answer to a Question on Quora: Is there any idempotent matrix that is not normal?

I came up with exactly the same answer

$$v_1 = \left[\begin{array}{cr} 1 & -1 \\0 & 0 \end{array}\right]$$

as Alex Eustis did, by looking at perhaps one of the simplest possible examples: a $$2\times 2$$ matrix diagonalisable with eigenvalues 1 and 0 in the basis made of vectors

$$v_1 = \left[\begin{array}{c} 1 \\0 \end{array}\right] \; \mbox{ and }\; v_2= \left[\begin{array}{c} 1 \\1 \end{array}\right]$$

which are not orthogonal to each other. Can you suggest a simpler non-orthogonal basis in $$\mathbb{R}^2$$?