Suppose that T : V - V is an invertible diagonalizable linear operator. Show that T^-1 is also diagonalizable.

Question

Suppose that T : V -> V is an invertible diagonalizable linear operator. Show that T^-1 is also diagonalizable.

mathteacher1729 · Answer

If it's diagonalizable, that means you can write the matrix T as PDP inverse where D is a diagonal matrix with the eigenvalues of the transformation on its diagonal and P is an invertible matrix composed of the corresponding eigenvectors. 

I'll leave the rest to you. :)

anonymous · Answer

Thank you! I'll bother you if I have problems going from there

anonymous · Answer

I'm a bit stuck; I think I'm supposed to represent the matrix in some way with variables and relate that to the eigenvalues but I'm not even sure how to do that ><

mathteacher1729 · Answer

Awesome free text: http://joshua.smcvt.edu/linearalgebra/index.html And Paul's Online math notes: http://tutorial.math.lamar.edu/Classes/LinAlg/Diagonalization.aspx I'm signing off for the evening, I'm tired and it's bedtime. Hope this helps!