We employ the machinery of linear relations to the study of optimization problems in linear algebra. We first show that the relational version of the pseudo-inverse can be realized through a generalization of the least-squares problem. This allows one to prove that the pseudo-inverse realizes the solution of certain relational optimization problems. Our main result is showing that a certain truncation of this pseudo-inverse defines a solution to a relational version of the classical low-rank approximation problem which recovers both the Eckart-Young Theorem and several optimization problems involving pairs of matrices and vector spaces.

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