Wedge product of matrices

Question

Can anyone explain what is the wedge product of two matrices? It is denoted by $A \wedge B$ I tried everything but I still cannot figure it out. Thanks.

Answer 1

I don't see anyone answering, so I will throw out an educated guess at a possibility. If such a product is defined, I would think it would be from the definition of wedge product of vectors. thus $C=A \wedge B$ would be $$C_{ij} = A_{i*} \wedge B_{*j}^T$$ where $A_{i*}$ is the row $i$ vector from $A$ and $B_{*j}$ is the column vector $j$ from $B$.

WARNING: I have actually not heard of the wedge product until now, but I have heard of different type of definitions such as this, and it is the natural extension of an operation defined only for vectors.

Answer 2

Let $k$ be a commutative ring (for example a field).
Suppose a linear map $u:k^m\to k^n$ has matrix $A=(a_{ij})$ in the canonical bases $(e_i)$ and $(f_j)$ of $ k^m$ and $k^n$.
Then $A\wedge A=\wedge^2A$ is the matrix of $\wedge^2 u:\wedge^2 k^m \to \wedge^2 k^n$ in the canonical bases $(e_i\wedge e_j)_{i\lt j}$ of $\wedge^2 k^m$ and $(f_k\wedge f_l)_{ k\lt l}$ of $\wedge^2 k^n$.
And that matrix $\wedge^2A$ has entry $((i,j)(k,l))$ equal to the minor \begin{array}{|cc|} a_{ik} & a_{il} \\ a_{jk}& a_{jl} \\ \end{array}

I suppose that it is possible to generalize the above to the case of the wedge product $A\wedge B$ of two different matrices, but I have not checked it: caveat wedgeator !

Bibliography Bourbaki , Algebra, Ch.3, § 8.5, Prop.10.