COROLLARY  Suppose $\alpha = \beta$.  A
sufficient condition for the spectrum of $A$ to be real is
\[
\parallel B \parallel + 2\parallel \alpha \parallel \leq |t|.
\]

THEOREM  
Suppose that $\lambda \in \Lambda$; that is, $\lambda$ is a Lagrange
multiplier of $\mu(\cdot)$ with respect to $S_{n-1}$.  Then $x \in
S_{\mu}(\lambda)$ if and only if $\lambda$ is an eigenvalue of $A$ with
$t=\mu(x)$, in which case an associated eigenvector is
\[ \left(
\begin{array}{c}
x\\
1
\end{array}
\right). \]
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APPLICATIONS

exponential cone nonnegativity

cone decompositions

interlacing

majorization

inverse eigenvalue problems

eigenvalue bounds