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**Fair to say there are "twice" as many square matrices as rectangular?**

Is it fair to say that there are at least twice as many square matrices as there are rectangular?

I was thinking something like this....

Let R be a rectangular matrix with m rows and n columns, and suppose either m < n or m > n. Then, we can associate two square matrices with R, namely RRt, and RtR, with Rt being R Transpose.

In other words, for every rectangular matrix there can be associated (at least) two square matrices.

Google brought up nothing, so I figured I would ask it here. It's not for homework or anything; just out of interest.