Can somebody see whether my intuition re diagonalizability is correct?

Dec 2018
Tel Aviv
Hi guys, can anybody see whether my intuition is correct?

True or False?

Let A and B be matrices of n x n.

1. If A and B are diagonalizable and they have the same characteristic polynomial, then A and B are similar.
2. If A and B are row equivalent and A is diagonalizable, then B is diagonalizable.

My intuitive answer is "false" to 1, and "true" to 2.

However, I am not sure, and either way, I would ideally like to be able to prove it...

Many thanks!
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Sep 2016

1. If a matrix is diagonalizable it means its eigenvectors form a basis (show this). So both $A$ and $B$ have $n$ eigenvectors which you can order as $\{v_1,\dotsc,v_n\}$ and $\{w_1,\dotsc,w_n\}$ respectively. Let $C$ be the linear transformation which maps $v_i \mapsto w_i$ and note that $C$ must be invertible (prove this). This should be enough.

2. Row operations can (and should) be thought of as special linear transformations (sometimes called elementary operators). The important thing here is they are invertible. Now, $A$ is row equivalent to $B$ means you can factor as $A = E_kE_{k-1}\dots E_1B$ where $E_i$'s are elementary. Now apply both sides of this to an eigenvector for $A$ and see what happens.
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