Linear Algebra - Diagonalizable and Eigenvalue Proof

by B_Phoenix
Tags: algebra, diagonalizable, eigenvalue, linear, proof
 P: 1 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.
 P: 111 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 $$A = S D S^{-1}$$.

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