# Linear Algebra - Diagonalizable and Eigenvalue Proof

1. Apr 7, 2009

### B_Phoenix

1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

I had no idea where to start.

2. Apr 7, 2009

### 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 $$A = S D S^{-1}$$.