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determine if the following matrices are similar, if yes, prove it. \begin{pmatrix} 2 & 1 \\ 0 & 2 \\ \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 0 & 2 \\ \end{pmatrix}

I checked some shared properties between similar matrices, such as determinant, characteristic polynomial and trace. Everything seems fine, I guess they are similar, but how to prove this? Thank you in advance.

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Hint: What happens if you express the second matrix on another basis? – Marc van Leeuwen Apr 2 '13 at 16:44
up vote 3 down vote accepted

Note that the second matrix is $2I$. IF the two matrices are similar, then $\pmatrix{2&1\\ 0&2}=P(2I)P^{-1}$ for some invertible matrix $P$. Now, do you know what is $P(2I)P^{-1}$ equal to?

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yes, $2I$, got it. thank you – user63219 Apr 2 '13 at 16:44

They are not similiar, cause you can't diagonalize the first one, as you only find one eigenvector, but the second one is already diagonal.

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thank you so much – user63219 Apr 2 '13 at 16:42

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