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$R = S + xx^t$ where $x \in \mathbb{R}^n$ and $R$ and $S$ are $n \times n$ covariance matrices. Is there anything I can say about the eigenvectors of $R$ and $S$? Or at least the largest eigenvalue and eigenvector pair of R? Is there a way to express the eigenvectors of $R$ in terms of $S$ and $x$? |
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The rank one matrix $x x^t$ has a single nonzero eigenvalue $\lambda=|x|^2$, with $x$ itself as eigenvector, because $ (x x^t) x = |x|^2 x$. Now, unless $x$ happens to be an eigenvector of $S$, I dont' think you can say anything about the eigenvectors of $R$. Regarding the largest eigenvalue of $R$ (I guess you meant eigenvalue instead of eigenvector?), recall that $$\lambda^{max}_R = max_{|w|=1} ( w^T R w)$$ but $$ w^T R w = w^T S w + w^T x x^t w \le \lambda^{max}_S + |x|^2$$ Further, calling $w_S$ the normalized eigenvector associated to $\lambda^{max}_S$, we have $$max_{|w|=1} ( w^T R w) \ge w_S^T R w_S = w_S^T S w_S + w_S^T x x^t w_S = \lambda^{max}_S + w_S^T x x^t w_S \ge \lambda^{max}_S $$ becase $w^T x x^t w \ge 0$ $\forall w$ Hence, I think that's about the only thing you can assert: $$ \lambda^{max}_S \le \lambda^{max}_R \le \lambda^{max}_S + |x|^2$$ |
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