Obtaining the relations of variables using diagonal matrices

Question

I know this is too an elementary question, but I am a pathetic newbie in mathematica and I cannot find any answer by googling.

What I want is obtaining some relations of unknown variables using their restrictions, which are given by

\begin{equation} \sum_{k}A_{ik}A_{jk}=0, \qquad A_{ij}=-A_{ji} \end{equation} where $(i,j,k)$ are from 1 to 6. These are, of course, represented as matrix forms by \begin{equation} AA^T=0, \qquad A^T=-A. \end{equation} And I need to obtain some relations of $A_{ij}$'s, e.g., $A_{25}A_{15}+A_{45}A_{35}$, from these matrix equation. Thank you for your advise.

Answer 1

Not exactly sure what you want. But check whether the following is what you want. For simplicity, I consider the case for the dimension 3, however you can simply change dim = 6 for your needs.

dim = 3;
mat = Array[a, {dim, dim}]; 
% // MatrixForm

mat2 = ReplacePart[ 
   mat, { {i_, i_} -> 0, {i_, j_} /; i > j  :>   - a[j, i]} ];
% // MatrixForm

# == 0 & /@ ( mat2. Transpose[mat2] // Flatten) // TableForm

The answer looks like

a[1,2]^2+a[1,3]^2==0
a[1,3] a[2,3]==0
-a[1,2] a[2,3]==0
a[1,3] a[2,3]==0
a[1,2]^2+a[2,3]^2==0
a[1,2] a[1,3]==0
-a[1,2] a[2,3]==0
a[1,2] a[1,3]==0
a[1,3]^2+a[2,3]^2==0

Answer 2

Try

A.A == 0
A == -Transpose[A]

See documentation for details.