Require square matrices in UpperHessenberg

[jishnub]

Since an UpperHessenberg matrix is square, we may require this condition in the constructor. Currently, this accepts a rectangular parent as well.

[codecov]

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  Coverage   93.73%   93.74%           
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+ Misses        987      986    -1     

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[stevengj]

I don't agree with this. It's not that unusual to talk about non-square upper-triangular matrices, so why not non-square upper-Hessenberg? The concept seems well-defined to me, and is currently supported:

julia> UpperHessenberg(rand(0:9, 7,4))
7×4 UpperHessenberg{Int64, Matrix{Int64}}:
 2  0  7  3
 8  9  0  4
 ⋅  5  0  4
 ⋅  ⋅  8  0
 ⋅  ⋅  ⋅  5
 ⋅  ⋅  ⋅  ⋅
 ⋅  ⋅  ⋅  ⋅

julia> UpperHessenberg(rand(0:9, 4,7))
4×7 UpperHessenberg{Int64, Matrix{Int64}}:
 6  3  6  8  4  8  8
 8  8  3  2  6  5  4
 ⋅  4  2  0  7  3  5
 ⋅  ⋅  6  2  8  9  6

Right now, there's not much you can do with a non-square upper-Hessenberg factorization, but there's an open issue for fast QR factorization (#625), which could be done in $O(m^2)$ time for an $m \times n$ upper-Hessenberg matrix using Givens rotations, and this could potentially be useful for Hessenberg least-square problems.

For example, GMRES is a famous case where non-square upper-Hessenberg least-square problems arise.

[jishnub]

Thank you. As long as there's a genuine use case, I'll close this