Eigen-values of $A$ and $B = A -kI$ [closed]
Let $A$, $B$ be $n\times n$ matrices and $B = A -kI$, where $k$ is a real number and $I$ is the $n\times n$ identity matrix. Then can we say that $A$ and $B$ have same eigenvectors?
My attempt: take $x$ which is eigenvalue of $A$ then $Au = xu$, and $(A- kI)u = Au -ku = (x-k)u = Bu$. So if $x$ is eigenvalue of $A$ with eigenvector $u$ then $x-k$ is eigenvalue of $B$ with corresponding eigenvector $u$.
Solution 1:
Yes, we can say that $A$ and $B$ have the same eigenvectors. The justification that you provided is correct.