- #1
AngrySaki
- 8
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The question is at the end of a chapter on spanning vector spaces.
Let P denote an invertible n x n matrix.
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 each n x n matrix A. [Here [tex]E_{\lambda}(A)}[/tex] is the set of eigenvectors of A.]
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
Homework Statement
Let P denote an invertible n x n matrix.
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 each n x n matrix A. [Here [tex]E_{\lambda}(A)}[/tex] is the set of eigenvectors of A.]
Homework Equations
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