Once upon a time (true story) a highly respected “pure” mathematician saw the following expression: $$ M = A(A^\top A)^{-1} A^\top $$ and instantly said it's the identity matrix. But if $A$ has many more rows than columns, then $M$ is actually a symmetric idempotent matrix of low rank, the mapping $x\mapsto Mx$ being the orthogonal projection onto the column space of $A.$
I have this suspicion: that “purity” of this sort is correlated with assuming matrices are square. And that this is only one instance of tacit simplifying assumptions correlated with “purity.”
(Paul Halmos, who was quite “pure,” wrote Finite-Dimensional Vector Spaces, in which the spectral decomposition of of Hermitian matrices is treated as important and (if I'm not mistaken) the singular-value decomposition, very important in applications, does not appear. So maybe that's another instance.)
Are there
- informed views about the merits or demerits of my suspicion; or
- other data points of interest?