Example with 2x2 orthogonal matrix
In[2]:= A = {{Cos[\[Theta]], -Sin[\[Theta]]}, {Sin[\[Theta]],
Cos[\[Theta]]}};
In[3]:= AT = Transpose[A];
In[4]:= Simplify[A.AT]
Out[4]= {{1, 0}, {0, 1}}
Now, in terms of summations
In[5]:= Simplify[Sum[A[[1, i]] AT[[i, 1]], {i, 1, 2}]]
Out[5]= 1
In[6]:= Simplify[Sum[A[[1, i]] AT[[i, 2]], {i, 1, 2}]]
Out[6]= 0
etc.
