Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This question is probably very elementary.

Basically I want to prove that conjugate matrices represent the same linear map but in different bases.

It is intuitively clear since if $M=X^{-1}NX$ and $N$ is expressed wrt basis $V$ then $X$ gives the coefficients of the linear combinations of vectors in $V$ for the new basis wrt which $M$ is expressed. So take a basis vector for $M$, then $X$ maps it to its representation wrt $V$ then $N$ acts on it then $X^{-1}$ maps it back to the expression wrt $M$'s basis.

But I don't know how to argue this rigorously. Maybe I also have to mention that $N$ is a linear map so this works? Please help!

Thanks.

share|improve this question

1 Answer 1

I think you did argue pretty formally, and perhaps I'd only add the following: any invertible operator can be seen as one mapping a basis into another basis (in fact, this can be taken as the definition of invertible operator: one that maps a basis into a basis) , so X in your post is representing an operator changing basis...

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.