Suppose you have a set of $n$ linearly independent vectors $v_1, v_2, ..., v_n$. Then we can call their wedge product $W = v_1 \wedge v_2 \wedge ... \wedge v_n$.

The $\ell_2$ norm $\|W\|_2$ is equal to the volume of the parallelotope generated by the vectors. The same value can be obtained in some different ways from the matrix $M = [v_1 | v_2 | ... | v_n]$ (assuming the $v_i$ are column vectors):

  • As the square root of the Gram determinant $|\det(M'\cdot M)|$
  • As the product of the singular values of $M$

I'm looking for a similar characterization of the $\ell_1$ norm $\|W\|_1$. Is there some easy way to compute this value from the matrix $M$, which is faster than computing the wedge product directly?


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