Distance between matrices with perturb eigenvectors

Question

I have two matrices A,B with eigenvectors $v_1(A),..,v_n(A)$ and $v_1(B),..,v_n(B)$ such that for all i $||(v_i(A)-v_i(B))||<\epsilon$.

Can I give a non-trivial bound for $||A-B||$?

Answer 1

Some initial steps that you might find helpful:

I assume that $\|\cdot\|$ refers to the Euclidean vector norm and the spectral norm for matrices.

First of all, we consider the case where $\lambda_i(A) = \lambda_i(B) =: \lambda_i$. Let $P_A$ denote the matrix with columns $v_1(A),\dots,v_n(A)$ and define $P_B$ similarly. Let $\Lambda$ be the diagonal matrix with diagonal entries $\lambda_1,\dots,\lambda_n$. Let $I$ denote the identity matrix. We note that $\|P_A - P_B\| \leq n \epsilon,$ and $$ \|P_A^{-1} - P_B^{-1}\| = \|P_A^{-1}(P_B - P_A)P_B^{-1}\| \leq \|P_A^{-1}\| \cdot \|P_B^{-1}\| \cdot \|P_A - P_B\|. $$

From there, we have \begin{align} \|A - B\| &= \|P_A\Lambda P_A^{-1} - P_B\Lambda P_B^{-1}\| \\ & \leq \|(P_A\Lambda P_A^{-1} - P_B\Lambda P_A^{-1})\| + \|(P_B\Lambda P_A^{-1} - P_B\Lambda P_B^{-1})\| \\ & =\|(P_A - P_B)\Lambda P_A^{-1}\| + \|P_B\Lambda(P_A^{-1} - P_B^{-1})\| \\ & \leq \|P_A - P_B\| \cdot \|\Lambda\| \cdot \|P_A^{-1}\| + \|P_B\|\cdot \|\Lambda\| \cdot \|P_A^{-1} - P_B^{-1}\| \\ & \leq \|P_A - P_B\| \cdot \|\Lambda\| \cdot \|P_A^{-1}\| + \|P_A - P_B\|\cdot \|\Lambda\|\cdot \|P_A^{-1}\| \cdot \|P_B^{-1}\|\cdot \|P_B\| \\ & = \|P_A - P_B\| \cdot \|\Lambda\| \cdot \|P_A^{-1}\|(1 + \|P_B^{-1}\|\cdot \|P_B\|) \\ & = \epsilon n \|\Lambda\| \cdot \|P_A^{-1}\|(1 + \kappa(P_B)), \end{align} where $\kappa(M)$ denotes the condition number of $M$.

From there, I'd look for an upper bound to $\kappa(P_B)$ in terms of $\kappa(P_A)$ and $\epsilon$.