The "north pole problem" and random orthogonal matrices
This paper is motivated by the following observation. Take a 3x3 random (Haar distributed) orthogonal matrix [Gamma], and use it to "rotate" the north pole, x0 say, on the unit sphere in R3. This then gives a point u=[Gamma]x0 that is uniformly distributed on the unit sphere. Now use the s...
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ftrepec:oai:RePEc:eee:stapro:v:79:y:2009:i:17:p:1878-1883 2024-04-14T08:16:21+00:00 The "north pole problem" and random orthogonal matrices Eaton, Morris L. Muirhead, Robb J. http://www.sciencedirect.com/science/article/pii/S0167-7152(09)00195-3 unknown http://www.sciencedirect.com/science/article/pii/S0167-7152(09)00195-3 article ftrepec 2024-03-19T10:31:50Z This paper is motivated by the following observation. Take a 3x3 random (Haar distributed) orthogonal matrix [Gamma], and use it to "rotate" the north pole, x0 say, on the unit sphere in R3. This then gives a point u=[Gamma]x0 that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving v=[Gamma]u=[Gamma]2x0. Simulations reported in Marzetta et al. [Marzetta, T.L., Hassibi, B., Hochwald, B.M., 2002. Structured unitary space-time autocoding constellations. IEEE Transactions on Information Theory 48 (4) 942-950] suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, moreover, that w=[Gamma]3x0 has higher probability of being closer to the poles ±x0 than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension p>=3, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in Rp, where p>=3. What are the distributions of U2=x'[Gamma]2x and U3=x'[Gamma]3x? It is clear by orthogonal invariance that these distributions do not depend on x, so that we can, without loss of generality, take x to be x0=(1,0,.,0)'[set membership, variant]Rp. Call this the "north pole". Then is the first component of the vector [Gamma]kx0. We derive stochastic representations for the exact distributions of U2 and U3 in terms of random variables with known distributions. Article in Journal/Newspaper North Pole RePEc (Research Papers in Economics) North Pole |
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RePEc (Research Papers in Economics) |
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This paper is motivated by the following observation. Take a 3x3 random (Haar distributed) orthogonal matrix [Gamma], and use it to "rotate" the north pole, x0 say, on the unit sphere in R3. This then gives a point u=[Gamma]x0 that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform u, giving v=[Gamma]u=[Gamma]2x0. Simulations reported in Marzetta et al. [Marzetta, T.L., Hassibi, B., Hochwald, B.M., 2002. Structured unitary space-time autocoding constellations. IEEE Transactions on Information Theory 48 (4) 942-950] suggest that v is more likely to be in the northern hemisphere than in the southern hemisphere, and, moreover, that w=[Gamma]3x0 has higher probability of being closer to the poles ±x0 than the uniformly distributed point u. In this paper we prove these results, in the general setting of dimension p>=3, by deriving the exact distributions of the relevant components of u and v. The essential questions answered are the following. Let x be any fixed point on the unit sphere in Rp, where p>=3. What are the distributions of U2=x'[Gamma]2x and U3=x'[Gamma]3x? It is clear by orthogonal invariance that these distributions do not depend on x, so that we can, without loss of generality, take x to be x0=(1,0,.,0)'[set membership, variant]Rp. Call this the "north pole". Then is the first component of the vector [Gamma]kx0. We derive stochastic representations for the exact distributions of U2 and U3 in terms of random variables with known distributions. |
format |
Article in Journal/Newspaper |
author |
Eaton, Morris L. Muirhead, Robb J. |
spellingShingle |
Eaton, Morris L. Muirhead, Robb J. The "north pole problem" and random orthogonal matrices |
author_facet |
Eaton, Morris L. Muirhead, Robb J. |
author_sort |
Eaton, Morris L. |
title |
The "north pole problem" and random orthogonal matrices |
title_short |
The "north pole problem" and random orthogonal matrices |
title_full |
The "north pole problem" and random orthogonal matrices |
title_fullStr |
The "north pole problem" and random orthogonal matrices |
title_full_unstemmed |
The "north pole problem" and random orthogonal matrices |
title_sort |
"north pole problem" and random orthogonal matrices |
url |
http://www.sciencedirect.com/science/article/pii/S0167-7152(09)00195-3 |
geographic |
North Pole |
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North Pole |
genre |
North Pole |
genre_facet |
North Pole |
op_relation |
http://www.sciencedirect.com/science/article/pii/S0167-7152(09)00195-3 |
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1796315010479161344 |