Formula for overdetermined logical matrix pseudoinverse not requiring SVD?

In https://commons.wikimedia.org/wiki/File:YI_%3D_PI.png, you will find a formula-based solution for an overdetermined logical matrix pseudoinverse. This simple formula gives the same result as the Moore-Penrose method. Does anyone know of any other overdetermined logical matrices that can be solved without using SVD methods?

• Your question is more likely to get an answer if it includes an example written here in proper mathematical notation, rather than as code in a language not everyone knows, with undefined variables in it, obfuscated behind an image and a link. Nov 6 '20 at 17:29
• Anyhow, you know that there is a closed formula for inverses of full-rank matrices that does not require the SVD, right? $(A^TA)^{-1}A^T$ or $A^T(AA^T)^{-1}$, depending on the dimensions. Nov 6 '20 at 17:31
• I am sorry about the Mathematica jargon. I have no idea how to use proper mathematical notation for computation where I use Tuples and Partition. Nov 6 '20 at 20:48
• Thank you for your additional comment. For a=4, w=3, the matrix is 64x12 and it will reduce in rank to 10. With row reduction, the matrix is neither logical or binary. Nov 6 '20 at 20:51
• It is very clear that I need a coauthor. I am doug@youvan.com . Retired, I no longer have access to graduate students and postdocs as in the past at MIT. Nov 6 '20 at 20:54