The question is at the end of a chapter on spanning vector spaces.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Let P denote an invertiblen x nmatrix.

If [tex]\lambda[/tex] is a number, show that

[tex]E_{\lambda}(PAP^{-1}) = \left\{PX | X\;is\;in\;E_{\lambda}(A)\right\}[/tex]

for eachn x nmatrix A. [Here [tex]E_{\lambda}(A)}[/tex] is the set of eigenvectors ofA.]

2. Relevant equations

3. The attempt at a solution

I'm having trouble understanding what the equality means, or how to read it.

The left side looks to be the eigenvectors of a diagonal matrix, which I think are always the columns of an identity matrix.

On the right side, I assume the matrix P would be the eigenvectors of A, so I think it's the span of the products of P multiplied by each of it's columns.

I don't know what to make of those ideas, so I think I'm either missing something about eigenvectors or spanning (very possible), or reading the question wrong.

Thanks

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# Homework Help: Eigenvectors & subspace spanning

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