# Operators $T_i$ in matrix form.

Let $$V$$ be a $$K$$-vector space and let $$T$$ be a $$K$$-linear map $$T: V \otimes V \to V \otimes V$$. Denote by $$T_i$$ the linear operator on $$V^{\otimes n} \to V^{\otimes n}$$ which acts on the $$i$$th and $$(i+1)$$th factors and acts trivially on other factors. I would like to understand the matrix form of $$T_i$$. For example, let $$V$$ be $$2$$ dimensional and $$e_1, e_2$$ be its standard basis. Suppose that with respect to $$e_1\otimes e_1, e_1 \otimes e_2, e_2 \otimes e_1, e_2 \otimes e_2$$, $$T$$ is \begin{align*} \left[ \begin {array}{cccc} \mu_{{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}&\mu_{ {1,4}}\\ \mu_{{2,1}}&\mu_{{2,2}}&\mu_{{2,3}}&\mu_{{2 ,4}}\\ \mu_{{3,1}}&\mu_{{3,2}}&\mu_{{3,3}}&\mu_{{3,4 }}\\ \mu_{{4,1}}&\mu_{{4,2}}&\mu_{{4,3}}&\mu_{{4,4}} \end {array} \right] . \end{align*} Let $$n=3$$. I obtained that, with respect to $$e_1 \otimes e_1 \otimes e_1, e_1 \otimes e_1 \otimes e_2, e_1 \otimes e_2 \otimes e_1, e_1 \otimes e_2 \otimes e_2, e_2 \otimes e_1 \otimes e_1, e_2 \otimes e_1 \otimes e_2, e_2 \otimes e_2 \otimes e_1, e_2 \otimes e_2 \otimes e_2,$$ \begin{align*} T_1 = \left[ \begin {array}{cccccccc} \mu_{{1,1}}&0&\mu_{{1,2}}&0&\mu_{{1,3 }}&0&\mu_{{1,4}}&0\\ 0&\mu_{{1,1}}&0&\mu_{{1,2}}&0& \mu_{{1,3}}&0&\mu_{{1,4}}\\ \mu_{{2,1}}&0&\mu_{{2,2} }&0&\mu_{{2,3}}&0&\mu_{{2,4}}&0\\ 0&\mu_{{2,1}}&0& \mu_{{2,2}}&0&\mu_{{2,3}}&0&\mu_{{2,4}}\\ \mu_{{3,1} }&0&\mu_{{3,2}}&0&\mu_{{3,3}}&0&\mu_{{3,4}}&0\\ 0& \mu_{{3,1}}&0&\mu_{{3,2}}&0&\mu_{{3,3}}&0&\mu_{{3,4}} \\ \mu_{{4,1}}&0&\mu_{{4,2}}&0&\mu_{{4,3}}&0&\mu_{{4 ,4}}&0\\ 0&\mu_{{4,1}}&0&\mu_{{4,2}}&0&\mu_{{4,3}}&0 &\mu_{{4,4}}\end {array} \right], \end{align*} \begin{align*} T_2 = \left[ \begin {array}{cccccccc} \mu_{{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}& \mu_{{1,4}}&0&0&0&0\\ \mu_{{2,1}}&\mu_{{2,2}}&\mu_{{ 2,3}}&\mu_{{2,4}}&0&0&0&0\\ \mu_{{3,1}}&\mu_{{3,2}}& \mu_{{3,3}}&\mu_{{3,4}}&0&0&0&0\\ \mu_{{4,1}}&\mu_{{ 4,2}}&\mu_{{4,3}}&\mu_{{4,4}}&0&0&0&0\\ 0&0&0&0&\mu_ {{1,1}}&\mu_{{1,2}}&\mu_{{1,3}}&\mu_{{1,4}}\\ 0&0&0&0 &\mu_{{2,1}}&\mu_{{2,2}}&\mu_{{2,3}}&\mu_{{2,4}}\\ 0 &0&0&0&\mu_{{3,1}}&\mu_{{3,2}}&\mu_{{3,3}}&\mu_{{3,4}} \\ 0&0&0&0&\mu_{{4,1}}&\mu_{{4,2}}&\mu_{{4,3}}&\mu_{ {4,4}}\end {array} \right]. \end{align*} Are these computations correct? Thank you very much.

## 1 Answer

Your computations indeed appear to be correct. Your results for $$T_1,T_2$$ can be confirmed using the Kronecker product, since we have $$T_2 = I_2 \otimes T, \quad T_1 = T \otimes I_2.$$