# $\ell_1$ norm of multivector in exterior algebra

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?