The discussion centers on proving that two matrices, A and C-1AC, have the same eigenvalues when C is an invertible matrix. The key equation derived is B = C-1AC, leading to the determinant relationship det(A - λI) = det(B - λI). The solution involves manipulating determinants and recognizing that the matrices A and C do not necessarily commute, which is crucial for the proof.
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muzziMsyed21 said:Homework Statement
Let A and C be nxn matrices with C invertible. Prove that A and C-1AC have the same eigenvalues.
Homework Equations
B=C-1AC
The Attempt at a Solution
det(A-λI) =det(B-λI)
det(A-λI) =det(C-1AC - λI)
det(A-λI) =det(C-1AC - λC-1IC)
det(A-λI) =det[CC-1(A-λI)] <<<< can you factor out a CC-1??