Uniones de Hecho

irreducible decomposition (As in Equation 34). Then for any invertible element R_∈Int_{[X ;Y]}, there exists an (efficiently computable) orthogonal matrix T such that the matrix T_·R is an element of

Int_{[X ;Y]}and is orthogonal.

Proof Lemma 37 and Equation 36 together imply that the matrix R_·RT can always be written in the form

R_·RT=_⊕ν(Mz_ν⊗Id_ν)

Since R_·RT is symmetric, each of the matrices Mzν is also symmetric and must therefore possess

an orthogonal basis of eigenvectors. Define the matrix Szν to be the matrix whose columns are the

eigenvectors of Mzν.

1. (ST·R)(ST·R)T _{is a diagonal matrix:}

Each column of S is an eigenvector of R_·RTby standard properties of the direct sum and Kro- necker product. Since each of the matrices, Szν, is orthogonal, the matrix S is also orthogonal.

We have:

(ST_·R)(ST_·R)T =ST_·R_·RT_·S, =S−1_·R_·RT_·S, =D,

where D is a diagonal matrix of eigenvalues of R·RT. 2. ST_·R_∈Int_{[X ;Y]}:

By Equation 36, a matrix is an element of ComYif and only if it takes the form⊕ν(Szν⊗Idν).

Since S can be written in the required form, so can ST. We see that ST _∈ComY, and by the

proof of Lemma 35, we see that ST_·R_∈Int[X ;Y].

Finally, setting T =D1/2_·ST makes the matrix T_·R orthogonal (and does not change the fact

that T_·R_∈Int[X ;Y]).

We see that the complexity of computing T is dominated by the eigenspace decomposition of Mzν, which is O z3ν

. Pseudocode for computing orthogonal intertwining operators is given Algorithm 6.

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