Similar Matrices Confer Equivalent Linear Transformations
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Apologies - I know there are some similar questions on this site, but I couldn't quite apply what they are saying here. Let's say I have the following matrices: $$A=begin{bmatrix}-1&6\-2&6end{bmatrix}; B=begin{bmatrix}1&2\-1&4end{bmatrix}$$ I conclude that they are similar (without finding $P$ such that $PAP^{-1} = B$ ) because both have eigenvalues 2,3 and thus will be similar to $D=begin{bmatrix}2&0\0&3end{bmatrix}$ . If we let $x=begin{bmatrix}2\3 end{bmatrix}$ , we have: $$Ax=begin{bmatrix}16\14end{bmatrix}; Bx=begin{bmatrix}8\10end{bmatrix}$$ I was told, however, that if two matrices are similar then they induce the same linear transformation. How can they do this if they map the same vector to a different vector? What is meant by 'same' here?