Let say we have a symmetric matrix $A \in \mathbb{R}^{n \times n}$ and we are interested in the calculation of the biggest eigenvalue $\lambda_\textrm{big}$. If the eigenvector $x_\textrm{big}$ of this biggest eigenvalue is present in a certain subspace $\mathcal{V}$ spanned by the columns of a orthonormal matrix $V$, $\lambda_\textrm{big}$ will also be an eigenvalue of the projected eigenvalueproblem $V'AV$.
Let's assume that an approximation $\hat{x}_\textrm{big}$ of $x_\textrm{big}$ is present in our subspace $\mathcal{V}$. Then $V'AV$ will have an eigenvalue $\hat{\lambda}_\textrm{big}$ that is close to $\lambda_\textrm{big}$.
If we now extend the subspace $\mathcal{V}$ to $\mathcal{V}_\textrm{ext}$ wich is spanned by the columns of orthonormal columns $V_\textrm{ext} := [V \quad V_1]$. Then we now by the Cauchy's interlace theorem see on https://en.wikipedia.org/wiki/Min-max_theorem that there is an eigenvalue $\hat{\lambda}_\textrm{ext}$ of $V_\textrm{ext}'AV_\textrm{ext}$ such that $\hat{\lambda}_\textrm{big} \leq \hat{\lambda}_\textrm{ext} \leq \lambda_\textrm{big}$. So this means we have a better approximation of the biggest eigenvalue if we extend the subspace.
I am wondering if it is possible to do something familiar for the residual. The vector $V' \hat{x}_\textrm{big}$ is the approximation of the eigenvector $x_\textrm{big}$ in the subspace $\mathcal{V}$, so we can define the residual as $r_\mathcal{V} = A V' \hat{x}_\textrm{big} - \hat{\lambda}_\textrm{big} V' \hat{x}_\textrm{big}$. We can define in the same way $r_{\mathcal{V}_\textrm{ext}}$. It is not in general true that $\left \| r_{\mathcal{V}_\textrm{ext}} \right \| < \left \| r_{\mathcal{V}} \right \|$, my question is in which cases this is true and in which cases not? Can anyone point me to some theorems that deal with such things?