# decomposition of an orthogonal matrix

Hi,

I have a matrix : $W=I+U^TV$

• $dim(W)=(D,D)$
• $dim(U)=dim(V)=(N,D)$ with $N < < D$

I need it to be orthogonal ie $W^TW=I$

which gives me : $V^TU+U^TV+V^TUU^TV=0$

From that point, i don't know where to go. Have anyone got some ideas about that issue ?

Cheers

Guillaume

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I assume your second term should be $U^T V$. –  Jeff Schenker Feb 25 '11 at 17:14
Yes, i corrected it –  GuillaumeThomas Feb 25 '11 at 17:26

Do you want a general solution or simply a good family of examples? If you want examples, let $V=-U=\sqrt{2} S^T$ where $S$ is an isometry from $\mathbb{R}^N$ into $\mathbb{R}^D$.
Given $U$ and if it is full rank, we can choose $$V = -2 \left(UU^T \right)^{-1}U$$ and the matrix $$W = I - 2 U^T \left(UU^T \right)^{-1}U$$ is an orthonormal matrix. A special case of this is provided by Jeff.