# find symmetric ratio matrix

I am quite new here as well as into the field of linear algebra. Therefore, I would appreciate any help. The question that I would like to ask is:

Imagine that I have two matrices AA and BB, where AA is a bases matrix (m×rm \times r) and BB is the weights/coefficients matrix (r×nr \times n), A×B≈MA \times B \approx M (MM is my input data matrix). I am calculating now the ratio of the weights matrix by extracting the consistent matrix of B as follows:

R=(w1/w1w1/w2w1/w3⋯w1/wnw2/w1w2/w2w2/w3⋯w2/wnw3/w1w3/w2w3/w3⋯w3/wn⋮⋮⋮⋱wn/w1wn/w2wn/w3wn/wn)R =
\begin{pmatrix}
w_{1}/w_{1} & w_{1}/w_{2} & w_{1}/w_{3} & \cdots & w_{1}/w_{n} \\
w_{2}/w_{1} & w_{2}/w_{2} & w_{2}/w_{3} & \cdots & w_{2}/w_{n} \\
w_{3}/w_{1} & w_{3}/w_{2} & w_{3}/w_{3} & \cdots & w_{3}/w_{n} \\
\vdots & \vdots & \vdots & \ddots & \\
w_{n}/w_{1} & w_{n}/w_{2} & w_{n}/w_{3} & &w_{n}/w_{n}
\end{pmatrix}

Now I would like to re-calculate AA and BB (actually if I understand it correctly I need only to find BB and then re-weight my AA according to the new RR) such that my RR matrix (that will be extracted from the new BB) is a block symmetric matrix. So, the ratio values R_{12}\approx R_{21}, R_{13} \approx R_{31}, R_{23} \approx R_{32}R_{12}\approx R_{21}, R_{13} \approx R_{31}, R_{23} \approx R_{32} and so on…

Does anyone have an idea how I can apply this. I guess it should be something with equality constraints but I am not sure how to implement this. Thanks in advance, I would be glad to get some feedback.

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