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Let $A_1 \in \mathbb{R}^{k\times k}$ be a symmetric and positive definite matrix. Let $x_1,x_2 \in \mathbb{R}^k$. Suppose I have the outer product $x_1 x_1^T$, from this post we know that $x_1$ is an eigenvector of $x_1 x_1^T$ and $||x_1||^2$ is its only non-zero eigenvalue.

We also know that if we apply $A_1 x_1$ then $x_1$ is projected onto the eigenvectors of $A_1$ and scaled by their corresponding eigenvalues given by the symmetric decomposition i.e $A_1 = \sum_i \lambda_i u_i u_i^T $ where the $\lambda_i, u_i$ are one of the $k$ eigenvalues and eigenvectors corresponding to $A_1$.

Is it possible to relate the eigenvalues and eigenvectors of $(A_1x_1 )(A_1x_1)^T + (A_1x_2)( A_1x_2)^T $ to $x_1 x_1^T + x_2 x_2^T $. I am trying to study the effect of $A_1$ on the eigenvalues and eigenvectors of the sum of the outer products.

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