While reading "Algebraic Graph Theory", 2nd edition, by N. Biggs (ISBN 0521458978) the author presents a proof of his lemma 8.6 on page 56 and uses a fact which he defers to another publication ("Linear transformations in n-dimensional vector space: An introduction to the theory of Hilbert space" by H. Hamburger) which I do not have access to.
For a real matrix $M$ denote $ev(M)$ as the set of eigenvalues of $M$.
Let $n,r,s \in \mathbb N$ be positive natural numbers with $n = r + s$.
The above mentioned fact is as follows: If $A \in \mathbb R^{n,n}$ is symmetric and positive definite with $A = \begin{pmatrix} R & Q \\ Q^\intercal & S \\ \end{pmatrix}$ for $Q \in \mathbb R^{r,s}$ and $R^\intercal = R \in \mathbb R^{r,r}$ and $S^\intercal = S \in \mathbb R^{s,s}$, then $\max(ev(A)) \leq \max(ev(R)) + \max(ev(S))$.
Can someone provide an accessible solution?
We adopt all your variables from above! The solution is a bit technical. First some notions and notation:
1) For any matrix (linear map) $M \in \mathbb R^{n,n}$ let $rg(M)$ be the range of $M$.
2) Let $I_m \in \mathbb R^{m,m}$ be the identity matrix for positive $m \in \mathbb N$.
3) For $u,v,w \in \mathbb R^m$ define $\left\langle v,w\right\rangle := \sum_{\ell=1}^m v_\ell\cdot w_\ell$ as the standard scalar product on $\mathbb R^m$ and $||u||_2 = \sqrt{\left\langle u,u\right\rangle}$ as the associated norm.
Define the (symmetric) projections $P_R = \begin{pmatrix} I_r & 0 \\ 0 & 0 \\ \end{pmatrix} \in \mathbb R^{n,n}$ and $P_S = \begin{pmatrix} 0 & 0 \\ 0 & I_s \\ \end{pmatrix} \in \mathbb R^{n,n}$.
Consider and accept the following facts:
1) $\mathbb R^n = rg(P_R) \oplus rg(P_S)$ and $\forall x \in rg(P_R): P_R\circ x = x$ and $\forall y \in rg(P_S): P_S\circ y = y$
2) For every $u \in \mathbb R^n$ with $||u||_2 = 1$ there exist $w_R \in rg(P_R)$ with $||w_R||_2 = 1$ and $w_S \in rg(P_S)$ with $||w_S||_2 = 1$ and real numbers $c_R,c_S \in \mathbb R$ such that $u = c_R\cdot w_R + c_S\cdot w_S$ and $1 = c_R^2 + c_S^2$.
3) By a result of Courant-Fischer there is an eigenvector $u_0 \in \mathbb R^n$ with $||u_0||_2 = 1$ for the greatest eigenvalue of $A$ such that $e_0 := \max(ev(A)) = \left\langle A\circ u_0,u_0\right\rangle$.
4) Let $a,c \in \mathbb R_+$ and $b \in \mathbb R$ such that $\forall t \in \mathbb R: \psi(t) = a \cdot t^2 + 2 \cdot b \cdot t + c \geq 0$. Then (by curve sketching ) we can infer that $\forall t \in \mathbb R: \psi(t) \leq (1 + t^2)\cdot (a+c)$.
We begin the solution with the following definitons:
$A_R := P_R \circ A \circ P_R = \begin{pmatrix} R & 0 \\ 0 & 0 \\ \end{pmatrix} \in \mathbb R^{n,n}$
$A_S := P_S \circ A \circ P_S = \begin{pmatrix} 0 & 0 \\ 0 & S \\ \end{pmatrix} \in \mathbb R^{n,n}$
Find $w_R' \in rg(P_R)$ with $||w_R'||_2 = 1$ and $w_S' \in rg(P_S)$ with $||w_S'||_2 = 1$ and real numbers $c_R',c_S' \in \mathbb R$ such that $u_0 = c_R'\cdot w_R' + c_S'\cdot w_S'$ and $1 = c_R'^2 + c_S'^2$.
Without loss of generality we assume $c_S' \not= 0$.
Let $t_0 := \frac{c_R'}{c_S'}$ and $a := \left\langle A\circ w_R', w_R'\right\rangle$ and $c := \left\langle A\circ w_S', w_S'\right\rangle$ and $b := \left\langle A\circ w_R',w_S'\right\rangle = \left\langle w_S',A\circ w_R'\right\rangle = \left\langle A\circ w_S',w_R'\right\rangle$.
So, $\max(ev(A)) = \left\langle A\circ u_0,u_0\right\rangle = \left\langle A\circ (c_R'\cdot w_R' + c_S'\cdot w_S'),c_R'\cdot w_R' + c_S'\cdot w_S'\right\rangle = c_R'^2\cdot \left\langle A\circ w_R', w_R'\right\rangle + c_R'\cdot c_S'\cdot \left\langle A\circ w_R',w_S'\right\rangle + c_S'\cdot c_R'\cdot \left\langle A\circ w_S',w_R' \right\rangle + c_S'^2\cdot \left\langle A\circ w_S', w_S'\right\rangle = c_R'^2\cdot a + 2 \cdot c_R'\cdot c_S'\cdot b + c_S'^2\cdot c = c_S'^2\cdot (t_0^2 \cdot a + 2\cdot b \cdot t_0 + c) \leq c_S'^2\cdot(t_0^2+1)\cdot (a+c) = c_S'^2\cdot(\left(\frac{c_R'}{c_S'}\right)^2+1)\cdot (a+c) = (c_R'^2 + c_S'^2)\cdot (a+c) = 1\cdot (a+c) = \left\langle A\circ w_R', w_R'\right\rangle + \left\langle A\circ w_S', w_S'\right\rangle = \left\langle A\circ P_R\circ w_R', P_R\circ w_R'\right\rangle + \left\langle A\circ P_S\circ w_S', P_S\circ w_S'\right\rangle = \left\langle P_R\circ A\circ P_R\circ w_R', w_R'\right\rangle + \left\langle P_S\circ A\circ P_S\circ w_S', w_S'\right\rangle = \left\langle A_R\circ w_R', w_R'\right\rangle + \left\langle A_S\circ w_S', w_S'\right\rangle \leq \max(\{\left\langle A_R\circ x_R, x_R\right\rangle:: x_R \in rg(P_R) \wedge ||x_R||_2 = 1\}) + \max(\{\left\langle A_S\circ y_S, y_S\right\rangle:: y_S \in rg(P_S) \wedge ||y_S||_2 = 1\}) = \max(ev(A_R)) + \max(ev(A_S)) = \max(ev(R)) + \max(ev(S))$