# Dot product over upper matrix and matrix of units

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?

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