Suppose $A$ is an $n \times m$ matrix that represents a transformation from an $n$ dimensional vector space $V$ with basis $\{e_1, e_2, \cdots, e_n\}$ to an $m$ dimensional vector space $W$ with basis $\{f_1, f_2, \cdots, f_m\}$. I have the corresponding exterior algebras:

$\wedge^2(V)$ with basis $\beta_v = \{e_{i_1}\wedge e_{i_2} | 1 \leq i_1 < i_2 \leq n\}$

$\wedge^2(W)$ with basis $\beta_w = \{f_{j_1}\wedge f_{j_2} | 1 \leq j_1 < j_2 \leq m\}$

Define $\wedge^2(A) : \wedge^2(V) \rightarrow \wedge^2(W)$ by $\wedge^2(A)(v_1 \wedge v_2) = Av_1 \wedge Av_2$

Find a matrix representation of $\wedge^2(A)$ with respect to the bases $\beta_v$ and $\beta_w$.

This has been giving me a VERY HARD time today. I know the general procedure and it should be simple, but the book keeping is just giving me a very hard time and I am fatigued. I'm hoping there is some simple way.


Let $A$ be given by the matrix $(a_{i,j})$. The matrix of $\wedge^2 A$ is given by what used to be known as the second compound matrix $A^{(2)}$ of $A$. The rows/columns of $A^{(2)}$ are indexed by pairs $(i_1,i_2)$ with $1\le i_1<i_2\le n$. The entry in row $(i_1,i_2)$ and column $(j_1,j_2)$ is the determinant $$\left|\matrix{a_{i_1,j_1}&a_{i_1,j_2}\\a_{i_2,j_1}&a_{i_2,j_2}}\right|.$$

Of course this all extends to higher exterior powers.


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