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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?

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  • $\begingroup$ What is the matrix of all units? $\endgroup$ – José Carlos Santos Mar 25 '18 at 14:53
  • $\begingroup$ $$\begin{pmatrix} 1 & 1& ... & 1 \\ 1 &1 & ... &1 \\ ... & ... & ... & ...\\ 1 &1 & ... &1 \\ \end{pmatrix}$$ $\endgroup$ – Eugene Mar 25 '18 at 15:24

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