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Given some vectors, how many dimensions do you need to add (to their span) before you can find some mutually orthogonal vectors that project down to the original ones?

Or, more formally...

Suppose $v_1,v_2,\ldots,v_k \in \mathbb C^m$. Take $n \ge m$ as small as possible such that if we consider $\mathbb C^m$ as a subspace of $\mathbb C^n$ in the natural way, then there is a projection $\pi$ of $\mathbb C^n$ onto $\mathbb C^m$ such that for some mutually orthogonal vectors $\hat v_1,\hat v_2,\ldots,\hat v_n v_k \in \mathbb C^n$, $\pi(\hat v_i) = v_i$ for each $i$.

Intuitively, this $n$ provides a measure of how "far from orthogonal" the original vectors are. My (deliberately open-ended) question is the following. Does anyone recognize this $n$, or does the idea seem familiar to anyone from any other context? I can't identify it with or connect it to anything else I've encountered, but I'm wondering if it might appear in some other guise in linear algebra, or elsewhere.

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