Induced Action of Matrix on Tensor Product

Question

I've been asked to write the induced action of a matrix in $M_4(\mathbb{R})$ on $\mathbb{R}^4 \otimes \mathbb R^4$, but this terminology is unfamiliar to me. What does it mean for a matrix to induce an action on a tensor product?

Similarly, what does it mean for a matrix to induce an action on $\wedge^2 (\mathbb R^4)$?

score 1 · Answer 1 · answered Oct 06 '14 at 05:15

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In general, given $T:U\to U$ and $S:V\to V$, we have $T\otimes S:U\otimes V\to U\otimes V$, given by $u\otimes v\mapsto Tu\otimes Sv$.

This answers both questions, as $\wedge^2(\mathbb{R}^4)$ is a quotient of $\mathbb{R}^4\otimes \mathbb{R}^4$.

answered Oct 06 '14 at 05:15

Amitai Yuval

19,308

I've only been given a single matrix, though. Can a single matrix $T$ act on a tensor product $V \otimes V$? – Empiromancer Oct 06 '14 at 05:19
Of course it can. It only means that the above $U$ and $V$ are the same space, and $T$,$S$ are the same linear map. – Amitai Yuval Oct 06 '14 at 06:31

Induced Action of Matrix on Tensor Product

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