Calculate the determinant of a multilinear operator

Question

How to calculate the determinant of a multilinear operator? Is it something different from the determinant of the linear operator? Thanks.

Answer 1

As far as I'm aware, it doesn't make sense to talk about the determinant of a multilinear operator. The usual determinant requires that the linear operator "start" and "end" at the same vector space: specifically, given a linear operator $T\colon V\to V$, where $V$ is an $n$-dimensional vector space, applying the $n$th exterior power functor $\Lambda^n$ produces a new linear operator $(\Lambda^nT)\colon \Lambda^n V\to\Lambda^nV$. Because $\Lambda^nV$ is a one-dimensional vector space, any linear operator from it to itself is necessarily given by multiplication by some scalar, and this scalar is the determinant of $T$. Observe that if we had started with a linear operator $T$ starting at a vector space $V$ and ending at a different vector space $W$, it would be impossible to identify this scalar; if $W$ is of a different dimension than $V$, there's no hope, because $\Lambda^n V$ and $\Lambda^n W$ will be of different dimensions, but even if $W$ is of the same dimension as $V$, it would be necessary to choose bases of $V$ and $W$ so that they (and hence also their exterior powers) could be identified in a consistent way.

In contrast, an $m$-linear operator (for $m>1$) looks like $$V_1\times\cdots\times V_m\to W.$$ for some vector spaces $V_1,\ldots,V_m,W$, and the above process wouldn't be applicable.