Linear Algebra - Diagonalizable and Eigenvalue Proof

[B_Phoenix]

Homework Statement

"Let A be a diagonalizable n by n matrix. Show that if the multiplicity of an eigenvalue lambda is n, then A = lambda i"

Homework Equations

The Attempt at a Solution

I had no idea where to start.

 

[VKint]

Since A is diagonalizable, we can choose some invertible matrix S such that A = S D S^{-1}, where D is diagonal and the diagonal entries of D are the eigenvalues of A. We can translate the assumption regarding the multiplicity of \lambda into a statement about D, after which the result follows by using A = S D S^{-1}.