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

## Main Question or Discussion Point

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.

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mfb
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Every square matrix IS a rectangular matrix.

If you consider rectangular matrices which are not square matrices only:
For every square matrix S, I can produce an infinite set of rectangular matrices by writing the columns of S once, twice, three times, ... next to each other (like "SSSS" - not a multiplication!).
No, that argument does not work.

There is an infinite amount of matrices, both for square matrices and rectangular matrices (note that the former are a proper subset of the latter). Therefore, intuitive ways to compare their number break down. As another example: There are as many even integers as there are integers.

Whoops, I suppose a better way to have phrased my question was "Are there twice as many matrices whose dimensions are the same as those whose dimensions are different," but that's a great answer, thanks!

AlephZero
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