# Eigenvalues of special singular matrix

Let's consider a matrix $$A\in\mathbb{R}^{n\times n}$$, with eigenvalues $$\lambda_i$$. We assume $$A$$ is positive definite, so $$\lambda_i>0$$. Now lets consider the matrix

$$B =\lambda I - A,\qquad \lambda \in \lambda_i.$$ We know that $$B$$ is singular, even better we know $$\lambda_i$$ are intended to be found by computing for which $$\lambda\ \det(B)=0$$. Now let us denote $$\mu_i$$ to be the eigenvalues of $$B$$. We know that $$\mu_1=0$$, but I was wondering if we can say anyting about the rest of the eigenvalues of B.

I considered the following example:

$$A = \begin{pmatrix}14 & 38 & 26\\ 38 & 110 &94\\ 26 & 94 & 145\end{pmatrix}$$ With eigenvalues $$\lambda = [0.1879,\ 36.6743,\ 232.1378]$$. Then if we choos $$B = \lambda_1I-A$$, we get that $$\mu = [0,\ -36.6743,\ -232.1378] = \lambda_1-\lambda.$$

Now my question is if A) the statement $$\mu = \lambda_i -\lambda$$ holds for any matrix $$A$$ and all eigenvalues $$\lambda_i$$ and B) if this only holds for eigenvalues $$\lambda_i$$, or holds for all matrices structured as $$B = cI-A$$, where $$c$$ can be any scalar value and this is actually a known property.

Suppose that $$\mu$$ is an eigenvalue of $$B$$ and that $$B=c\operatorname{Id}-A$$ for some matrix $$A$$ and some number $$c$$. Then there is a non-zero vector $$v$$ such that $$B.v=\mu v$$. But\begin{align}B.v=\mu v&\iff(c\operatorname{Id}-A).v=\mu v\\&\iff cv-A.v=\mu v\\&\iff A.v=cv-\mu v\\&\iff A.v=(c-\mu)v.\end{align}So, whenever $$\mu$$ is an eigenvalue of $$B$$, $$c-\mu$$ is an eigenvalue of $$A$$. By the same argument, if $$\lambda$$ is an eigenvalue of $$A$$, $$c-\lambda$$ is an eigenvalue of $$B$$. Note that the only thing I need for this to work is that $$B$$ is a square matrix. Being symmetric is not relevant.