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eigenvalues of matrices or operators

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Eigendecomposition of a Gram matrix with specified columns

My question is this: Given an eigendecomposition $L = \sum_{n=1}^N \lambda_n v_n v_n^T$ of the likelihood kernel, can I write the eigenvalues and eigenvectors in terms of the components $q$ and $\phi$ … I am inclined to think this is not enough information to represent the eigenvalues and eigenvectors in terms of $q$ and $\phi$. Can anyone confirm this? …
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