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I would like to ask if somebody can help me with the following problem. I would like to generate synthetic data for dimension reduction algorithm testing. Specifically, I would like to have for example a matrix with 1000 rows (=points) and 50 columns (=features) but the real dimension of such matrix after dimension reduction should have only 10 features. How can I generate such matrix? I prefer python code but any advice helps me.

I found out that I can multiply a random matrix (with 1000 rows and 10 columns) by the transposed first matrix from SVD decomposition (with 10 rows and 50 columns) but I do not know why.

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