Maths - Orthogonal Matrices Reorthogonalising

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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