Same for the the orthogonal matrices, or the diagonal matrices, or the symmetric matrices, or the unit determinant matrices, or the singular matrices ... They are all sets of Lebesgue measure zero.
Orthogonal, diagonal, symmetric, and unit-determinant matrices are all sub-groups though, which makes them 'more special' then all shearing matrices.
Singular matrices are special in the sense that they keep the matrix monoid from being a group. My category theory isn't strong enough to characterize it, but this probably also has a name.
Edit: I think the singular matrices are the 'kernel' of the right adjoint of the forgetful functor from the category of groups to the category of monoids.
Though I must admit a lot of that sentence is my stringing together words I only vaguely know.
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Notably though, shearings are very 'rare'. Any pertubation will make a shearing no longer a shearing. At least, if I remember correctly.
Same for the the orthogonal matrices, or the diagonal matrices, or the symmetric matrices, or the unit determinant matrices, or the singular matrices ... They are all sets of Lebesgue measure zero.
Orthogonal, diagonal, symmetric, and unit-determinant matrices are all sub-groups though, which makes them 'more special' then all shearing matrices.
Singular matrices are special in the sense that they keep the matrix monoid from being a group. My category theory isn't strong enough to characterize it, but this probably also has a name.
Edit: I think the singular matrices are the 'kernel' of the right adjoint of the forgetful functor from the category of groups to the category of monoids. Though I must admit a lot of that sentence is my stringing together words I only vaguely know.