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Prove that if A and B are similar matrices, the dim $Null(A)$ = dim $Null(B)$

I'm not really sure where to start for this problem. Any help would be appreciated. Thanks

Ian Murphy
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  • Let $V \subseteq \mathbb{R}^n$ be $Null(A)$ and $W \subseteq \mathbb{R}^n$ be $Null(B)$. I know that there is an invertible matrix $P$ so that $B = PAP^{-1}$. Can you use $P$ to relate $V$ and $W$? – Zach L. Mar 25 '15 at 20:00

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Hint:

If $A=PBP^{-1}$, then for any $v \in null(A)$ , notice that $PBP^{-1}v=Av=0$, so that $BP^{-1}v=P^{-1}0=0$, so that $P^{-1}v \in null(B)$. Similarly, if $w \in null(B)$, then $Pw \in null(A)$.

Therefore the mapping $P^{-1}: null (A) \to null (B)$ is an isomorphism with inverse $P$.

shalop
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