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Muffins
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Homework Statement
Suppose A and B are similar matrices, and that (mu) is an eigenvalue of A. We know that (mu) is also an eigenvalue of B, with the same algebraic multiplicity(proved in class) Suppose that g is the geometric multiplicity of (mu), as an eigenvalue of B. Show that (mu) has geometric multiplicity g as an eigenvalue of A.
Homework Equations
A=S*B*S^-1
B=S^-1 *A*S
The Attempt at a Solution
We know that g = dim(EigenSpace of (mu) for (B)), so this tells us that the space has a basis
(v1, v2, ... , vg)
From our previously worked problem, we know that if A and B are similar, an v is an eigenvector for B, then S*v is an eigenvector for B, so it follows that
(Sv1, Sv2, ... , Svg) are eigenvectors for A. Now to verify that this is a basis of A, we need to show that they are linearly independent and span (EgienSpace of (mu) for A).
So, (c1*S*v1 + c2*S*v2 + ... + cg*S*vg) = 0 vector, factor out an S and we get
S*(c1*v1 + c2*v2 + ... + cg*vg) = 0 vector, and because the v's form a basis they're linearly independent, forcing the c's to be 0.
Here's where my troubles hit. I'm entirely confused as to how to show (S*v1, ... S*vg) span the eigenspace(mu) for A. My professor suggested to take an arbitrary vector w, which is an element of the eigenspace(mu) for A, and show that the w is a linear combination of the S*v's.. but I'm at a loss on how to logically show that.
Any help will be greatly appreciated guys!
Thank you mucho!