= Largest-eigenvalue interlacing for a principal submatrix
{title2=$\lambda_1\ge\mu_1\ge\lambda_2$}
Let a real <symmetric matrix> $M$ have ordered <eigenvalues> $\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_n$, with $n\ge2$. If $A$ is obtained by deleting one matching row and column, its largest eigenvalue $\mu_1$ satisfies
$$
\lambda_1\ge\mu_1\ge\lambda_2.
$$
The <Rayleigh quotient> of $A$ is that of $M$ restricted to vectors whose deleted coordinate is zero. This gives the upper bound. Some nonzero linear combination of the top two orthonormal <eigenvectors> has that coordinate zero; normalizing it gives a quotient at least $\lambda_2$. The argument does not require distinct eigenvalues.
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