Does there exist a dot product on space of matrix nxn (n>1) regarding to which the matrix of all units would be orthogonal to any upper triangular matrix?
First i thought to define dot product as $$(A,B) = \sum_{i,j=1}^n \hat \delta_{i,j} a_{i,j} b_{i,j}$$ where $$ \hat \delta_{i,j} = \begin{cases} 1, & \text{if $i\le i$} \\ 0, & \text{if $i \gt j$} \end{cases} $$
But when I try to prove that this IS a product dot there is a mistake: $(A,A)$ gives zero so my formula is not really a product dot. Any ideas how to find correct one please?