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# Offset Eigenvalues

LINALG-LHHKX2

If $A$ is an $n\times n$ complex matrix and $I$ is the $n\times n$ identity matrix and if $B=A+\rho I$ for $\rho\in\mathbb{C}$ then

A

$B$ has more eigenvalues than $A$ if $\rho>0$.

B

$B$ has fewer eigenvalues than $A$ if $\rho<0$.

C

$A$ and $B$ have the same eigenvalues for all $\rho\in\mathbb{R}$.

D

$A$ and $B$ have the same number of eigenvalues for all $\rho\in\mathbb{R}$.