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Homework Statement
Let A and B be similar matrices
a)Prove that A and B have the same eigenvalues
Homework Equations
None
The Attempt at a Solution
Firstly, i dont see how this can even be possible unless the matrices are exactly the same :S
Your way is too easy.Oh, I think that way is much too complicated!
Do it directly from the equation:
If [itex]Av= \lambda v[/itex] then, for any invertible P, [itex]P^{-1}Av= \lambda P^{-1}v[/itex]. Now define [itex]u= P^{-1}v[/itex].
what do u mean by the same characteristic equation?
Make that "The characteristic equation of matrix A is [tex] det(A- \lambda I) = 0. [/tex]"The the characteristic equation of matrix A is [tex] det(A- \lambda I). [/tex]
Sorry. What I should have said is that they have the same characteristic POLYNOMIAL.Make that "The characteristic equation of matrix A is [tex] det(A- \lambda I) = 0. [/tex]"
For it to be an equation, it at least has to have an equals sign.
They can, but it's not likely.ohhh i see...
since they can have the same eigenvalues, does this mean that the matrices can also have the same eigenvectors?