# Wedge product of matrices

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.

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Could you give some context? – martini Oct 12 '12 at 14:04
After learning a bit more about what the wedge product is, I believe it does not apply to two different matrices as in $A \wedge B$ (even though my answer described a method to extend any vector multiply to a matrix multiply). Does this agree with your findings? – adam W Mar 28 at 20:05

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.

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 I suppose this definition works only in $R^3$? – Pantelis Damianou Oct 12 '12 at 22:04 Not at all, it is a valid definition for any conformable matrix pair (only need same number of columns in $A$ as rows in $B$) – adam W Oct 12 '12 at 22:29

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.

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