2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.