Let $B$ be a symmetric, positive definite matrix and consider the problem
$$\begin{array}{ll} \text{maximize} & x^\top B x\\ \text{subject to} & \|x\| = 1\\ & b^\top x = 0\end{array}$$
for some unit vector $b$, not necessarily an eigenvector of $B$. If $b$ is an eigenvector, this is easy: just pick the largest eigenvalue among all eigenvectors orthogonal to $b$. But what if $b$ is not an eigenvector?
My intuition is as follows. Let $z_i$ be the eigenvectors of $B$ (with corresponding eigenvalues $\lambda_i$. Each eigenvector can be projected onto the orthogonal complement of $b$ by taking the vector rejection
$$\hat{b}_i = z_i – \left(b^\top z_i\right)b$$
I believe that the maximizer should be one of the $\hat{b}_i$ vectors, but I don't know how to prove it or how to further characterize the right $i$. I guess that it should depend on both $\lambda_i$ and $(b^\top z_i)^2$, but don't know how to proceed further. Would appreciate any suggestions.
Best Answer
One approach to the problem is as follows: extend $b$ into an orthonormal basis, so that $b,v_2,\dots,v_{n}$. Note that for any unit vector $y \in \Bbb R^{n-1}$, the matrix $Qy$ is a vector in $b^\perp$. Conversely, every unit vector $x \in b^\perp$ can be expressed as $Qy$ for some vector $y \in \Bbb R^{n-1}$.
So, we can reframe your question as $$ \max_{\|y\| = 1}(Qy)^TB(Qy) = \max_{\|y\| = 1}y^T (Q^TBQ)y. $$ In other words, the maximum that we want is the largest eigenvalue of $Q^TBQ$.
The Courant-Fischer theorem tells us that this maximum will necessarily fall between (or be equal to one of) the two largest eigenvalues of $B$.
It does not necessarily hold that the maximum is equal to the projection of the eigenvector. As an example, consider $$ B = \pmatrix{3&0&0\\ 0&2&0\\ 0&0&1}, \quad b = (1,1,1)/\sqrt{3} \implies\\ Q = \pmatrix{1/\sqrt{2}&1/\sqrt{6}\\-1/\sqrt{2}&1/\sqrt{6}\\0&-2/\sqrt{6}}, \quad Q^TBQ = \pmatrix{5/2 & 1/\sqrt{12}\\ 1/\sqrt{12} & 2/3}. $$ The maximizer here is $y = v/\|v\|$, with $v = (\frac 16 (11 \sqrt{3} + \sqrt{399}), 1)$. You can verify that $x = Qy$ is not the projection of $(1,0,0)$ onto $b^\perp$.