Maths - Reorthogonalising a matrix - Attempt at an Alternative Method

Another approach to try might be to split the matrices into there individual elements:

The inverse of a 2x2 matrix [O] is:

[O]-1 = 1/(o00 o11 * o01 o10)
o11 -o01
-o10 o00

so expanding out [O]T = [O]-1 gives:

o00 = o11/(o00 o11 * o01 o10)
o10 = -o01/(o00 o11 * o01 o10)
o01 = -o10/(o00 o11 * o01 o10)
o11 = o00/(o00 o11 * o01 o10)

If we constrain the determinant to be 1 then,

(o00 o11 * o01 o10) = 1

o00 = o11
o10 = -o01

we are looking for C where [O] = [C]*[M]

c00 * m00 + c01 * m10 c00 * m01 + c01 * m11
c10 * m00 + c11 * m10 c10 * m01 + c11 * m11
=
c00 c01
c10 c11
m00 m01
m10 m11

so combining these gives:

c00 * m00 + c01 * m10 = c10 * m01 + c11 * m11

c00 * m01 + c01 * m11 = -c10 * m00 - c11 * m10

There seem to be more unknowns than equations so its difficult to see where to take this?

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