Almost Complex
Let J be an nxn matrix with real entries, such that J^(2) = -I (where I is the nxn identity matrix).
Show that if n is odd then no such J exists and provide an example of such a J for every even n.
!Such a J is called a Linear Complex Structure https://en.wikipedia.org/wiki/Linear\_complex\_structure!<
Posted Jun 13, 2022, 9:00 PM UTC
120
a) no such odd-dimension example exists. Suppose n is odd, and J^2 = -I. Then det(J)^2 = det(-I) = (-1)^(n) det(I) = -1. So J has an imaginary determinant. But J has real values, so must have a real determinant. Contradiction!
b) consider the 2 by 2 example J = ((0,-1),(1,0)). Then J^2 = -I. We can generalize this to a 2n by 2n example with J = ((0,-I),(I,0)), where I is the n by n identity matrix.