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Given a diagonalizable matrix A prove or give a counter example:

$A^2$ and $A$ are row equivalent

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2 Answers 2

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The wiki page writes that

Because the null space of a matrix is the orthogonal complement of the row space, two matrices are row equivalent if and only if they have the same null space.

Assuming that $A$ is transformed into a basis in which it is already diagonal, its null space will be spanned by those basis vectors for which eigenvalue $0$ belongs, i.e. where there is $0$ in the diagonal. But $A^2$ will have $0$ at exactly the same diagonal places.

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Generally, when you have to prove something about a diagonalizable matrix, try changing the basis to one in which it is diagonal.

If $P^{-1}AP$ is diagonal, how does it compare to $P^{-1}A^2 P = (P^{-1}AP)^2$? If they are row equivalent (meaning you can multiply one on the left by an invertible matrix to get the other), can you do the same for $A$ and $A^2$?

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