# Definition of invariant tensor of type $(p,q)$

Prove that tensor of the type $$(1,1)$$ which is invariant under orthogonal transformations of $$\mathbb{R}^n$$ is proportional to the tensor $$\delta^{i}_{j}$$.

Approach: Let $$V=\mathbb{R}^n$$ and $$T:V\times V^*\to \mathbb{k}$$ be a tensor of type $$(1,1)$$. Let $$\{e_1,\dots,e_n\}$$ and $$\{\tilde{e}_1,\dots,\tilde{e}_n\}$$ be two basis of $$V$$ such that matrix $$C$$ is transformation matrix from $$(e)$$ to $$(\tilde{e})$$. Then $$\tilde{e}_j=c^l_je_l$$ and $$\tilde\varepsilon^i=d^i_k\varepsilon^k$$ where by $$\{\varepsilon^i\}$$ and $$\{\tilde{\varepsilon}^i\}$$ I mean corresponding dual basis.

Let $$\tilde{T}^i_j=T(\tilde{e}_j,\tilde{\varepsilon}^i)=T(c^l_je_l,d^i_k\varepsilon^k)=c^l_jd^i_kT(e_l,\varepsilon^k)=c^l_jd^i_kT^k_l$$, where $$c^i_j$$ and $$d^i_j$$ are elements of $$C$$ and $$C^{-1}$$, respectively.

I was wondering what does mean that our tensor is invariant under orthogonal transformation?

I have the following hypothesis: maybe $$T^i_j=c^l_jd^i_kT^k_l$$ for any orthogonal matrix $$C$$?

Note that in the LHS I wrote $$T^i_j$$ instead of $$\tilde{T}^i_j$$.

Am I wrong? Anyway what is the definition of invariant tensor? Would be very grateful for any comments!

EDIT: Actually I solved this problem before I posted my question so let me write down my solution: since our tensor of type $$(1,1)$$ is invariant then $$T^i_j=c^l_jd^i_kT^k_l$$ where $$c^i_j, d^i_j$$ are elements of matrix $$C$$ and $$C^{-1}$$, respectively and $$C$$ is orthogonal matrix.

Let's take the following diagonal matrix $$C_r$$ s.t. $$c_{rr}=-1$$ and $$c_{ii}=1$$ for $$i\neq r$$. We see that $$C_r$$ is orthogonal and $$C^{-1}=C$$. Then one can show that $$T^r_j=c^l_jd^r_kT^k_l=d^r_kT^k_jc^j_j=d^r_rT^r_jc^j_j$$ and for $$j\neq r$$ we have $$T^r_j=-T^r_j$$ and it means that $$T^r_j=0$$. Since $$r$$ is arbitrary then it follows that $$T^r_j=0$$ for all $$j\neq r$$.

Consider the following orthogonal transformation given by its matrix $$C$$ such that $$C$$ is obtained from identity matrix by interchanging $$i$$th and $$j$$th columns. Then $$T^i_i=c^l_id^i_kT^k_l=d^i_kT^k_jc^j_i=d^i_kT^k_j=T^j_jd^i_j=T^j_j$$

So it shows that $$T^i_j=c\delta^i_j$$ where $$c$$ is constant.

Write $$\widetilde{e}_j = A^i_{~j}e_i$$ for some orthogonal matrix $$(A^i_{~j})$$. Then the dual relations are $$\widetilde{e}^i = (A^\top)^i_{~j}e^j$$. The relation between the components of $$T$$ relative to both bases is $$\widetilde{T}^i_{~j} = (A^\top)^i_{~k}A^\ell_{~j}T^k_{~\ell},$$but the assumption for the problem is that $$T^i_{~j} = \widetilde{T}^i_{~j}$$. Hit both sides with $$A^p_{~i}$$ to get $$A^p_{~i}T^i_{~j} = A^p_{~i}(A^\top)^i_{~k}A^\ell_{~j}T^k_{~\ell} = \delta^p_{~k}A^\ell_{~j}T^k_{~\ell} = A^\ell_{~j}T^p_{~\ell}$$Because of this, the problem boils down, on the matrix level, to showing that if $$T$$ is a $$n\times n$$ matrix such that $$AT = TA$$ for all $$A \in {\rm O}(n,\Bbb R)$$, then $$T$$ is a multiple of $${\rm Id}_n$$. This, in turn, is explained in this post.
• Could you explain why $T^i_j=\tilde{T}_j^i$? I have asked you about my edit but you did not answer