Eigenvalues of a block matrix with all diagonal blocks but one

Let us consider a block matrix of the form

$$A= \begin{bmatrix} -(k+\mu)I & B \\ kI & -(\gamma + \mu)I \end{bmatrix},$$

where $$I$$ is the $$n\times n$$ identity matrix, $$\gamma, k$$ and $$\mu$$ are positive constants and

$$B= \begin{bmatrix} a_1b_1& \ldots & a_1b_n \\ \vdots& \ddots& \vdots \\ a_nb_1& \ldots& a_nb_n \end{bmatrix}$$

with $$a_i, b_i >0$$ for all $$i=1,\dots,n$$.

Which are the eigenvalues of matrix $$A$$? Is there any way to evaluate them through the eigenvalues of the blocks of matrix $$A$$?

• Hey :) It is good practice to include a bit of what you have tried in the question. Try to show where you got stuck. Jun 5 '20 at 11:43
• Sure, I'll edit it then, thanks for the advice! Jun 5 '20 at 12:53

Given vectors $$\rm{a}, \rm{b} \in \mathbb R^n$$, let

$$\rm{M} := \begin{bmatrix} -(\kappa + \mu) \rm{I}_n & \rm{a}\rm{b}^\top\\ \kappa \,\rm{I}_n & -(\gamma + \mu) \rm{I}_n \end{bmatrix}$$

whose characteristic polynomial is

\begin{aligned} \det \left( s \rm{I}_{2n} - \rm{M} \right) &= \det \begin{bmatrix} (s + \kappa + \mu) \rm{I}_n & - \rm{a}\rm{b}^\top\\ - \kappa \,\rm{I}_n & (s + \gamma + \mu) \rm{I}_n \end{bmatrix}\\ &= \det \left( (s + \kappa + \mu) (s + \gamma + \mu) \, \rm{I}_n - \kappa \, \rm{a}\rm{b}^\top \right)\end{aligned}

because multiples of the identity matrix do commute. Let $$q$$ be the characteristic polynomial of rank-$$1$$ matrix $$\kappa \, \rm{a}\rm{b}^\top$$. Hence,

$$q (s) = s^{n-1} \left( s - \kappa \, \rm{b}^\top \rm{a} \right)$$

and, thus,

$$\det \left( s \rm{I}_{2n} - \rm{M} \right) = q \left( (s + \kappa + \mu) (s + \gamma + \mu) \right)$$

• I think that the following is worth adding for the sake of completeness: the eigenvalues are the zeros of $\det(s I_{2n} - M)$, which in this case turn out to be $$\lambda = -(\kappa + \mu), -(\gamma + \mu),$$ each with algebraic mulitplicity $n-1$, and the solutions to the quadratic equation $$(\lambda + \kappa + \mu)(\lambda + \gamma + \mu) = \kappa b^Ta,$$ each with multiplicity $1$. Jun 5 '20 at 13:45