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Topic: Nx2N lapped orthogonal transform
Replies: 13   Last Post: Sep 12, 2013 7:38 PM

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Roland Franzius

Posts: 433
Registered: 12/7/04
Re: Nx2N lapped orthogonal transform
Posted: Aug 20, 2013 6:24 AM
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Am 19.08.2013 06:34, schrieb James Dow Allen:
> Let
> ( A B 0 )
> ( 0 A B )
> ( B 0 A )
> be a 3Nx3N real matrix with A,B,0 each NxN and 0 an all-zeros matrix.
>
> What is the necessary and sufficient condition for that matrix to be
> orthogonal, i.e. that its transpose also be its inverse?
>
> This problem statement can be considered ambiguous.
> *But if you derive a good parametric form for (A B) you will know it.*
>
> (I already "know the answer." I post from curiosity: Is this a VERY easy
> problem, or just an easy problem.)


Since orthogonality of a nxn real matrix X, X^+.X = Id is easier
expressed by "the columns as well as the rows of the matrix X form an
n-dimensional orthonormal frame:
X_i.X_k =delta_ik",
the condition is simply that the three 6d vectors

{a_i1,..a_i3, b_i1..b_i3}, i=1,2,3

form an orthonormal tripod in 6 dimensions.

Parametrization may be done in descriptive words by
"choose at random 3 different vectors from a unit vector in 6d spherical
coordinates {theta1..theta5,phi}, and partition that matrix in 2 x 3 x 3".

--

Roland Franzius





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