Imaginary matrices

Let's consider square matrices

The identity matrix is a diagonal from upper left to lower right of all
ones

The transpose matrix is, if I remember Wikipedia correctly gosh I'm sorry,
a single diagonal from upper right to lower left of all ones

Clearly the transpose of the transpose is identity making transpose the
second square root of the identity matrix

Let us consider now

The negative identity matrix with a diagonal from upper left to lower right
of all negative ones

Multiplying by this twice and gives identity so it is yet another square
root of the identity matrix

Consider now the negative transpose matrix with a diagonal from upper right
to lower left of all negative one s. Multiplying by this twice gives the
original matrix. So we see that the negative transpose matrix is yet
another square root of the identity matrix.


I wonder if these four square roots of unity

Can be extended to interpret the negative identity matrix as something
which itself might have a square root. My guess is the imaginary identity
matrix with a diagonal from upper left to lower right of all i, where I is
the square root of negative one, would suffice


Many thanks to John Baez to introducing me to the matrix multiplication
operation around a dozen years ago thanks again


[[Mod. note -- Wikipedia has lots of information about matrix square roots:
  https://en.wikipedia.org/wiki/Square_root_of_a_matrix
-- jt]]