## Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$

## Problem 378

Let $A$ be an $n \times n$ matrix and let $c$ be a complex number.

**(a)** For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to $\lambda+c$?

**(b)** Prove that the algebraic multiplicity of the eigenvalue $\lambda$ of $A$ is the same as the algebraic multiplicity of the eigenvalue $\lambda+c$ of $A+cI$ are equal.

**(c)** How about geometric multiplicities?