What are the eigenvalues of $A + a I, a \in \mathbb{R}$ when $A$ is real symmetric matrix?

Question

For example, let $A$ be a diagonal matrix, with eigenvalues $\lambda _1, \ldots, \lambda_n$

Then $\det(sI - (A+aI)) = \det((s-a)I - A)$ is a polynomial with roots $ a + \lambda_1$, $\ldots, a + \lambda_n.$

Hence the eigenvalues of $A + aI$ are $\lambda_i + a$

Does a similar result hold when $A$ is a real symmetric matrix?

Answer 1

This result even holds for all square matrices, not just symmetric ones. Consider the general eigenvalue equation $A v= \lambda v$ and $B = A + a I$. Then

$$Bv = (A + aI)v = Av + av = \lambda v + av = (\lambda + a) v,$$

hence the eigenvalues of $A + aI$ are $\lambda_i + a$, if the eigenvalues of $A$ are $\lambda_i$.