Versions of Cochran's theorem for general quadratic expressions in normal matrices
✍ Scribed by Tonghui Wang
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 523 KB
- Volume
- 58
- Category
- Article
- ISSN
- 0378-3758
No coin nor oath required. For personal study only.
✦ Synopsis
Let E, V be n-, p-dimensional inner product spaces over the real field, cg~( V, El be the set of all linear maps of V into E, and Y be a normal random operator in S(V, E) with mean /t and covariance L'> The necessary and sufficient conditions under which a set of general quadratic expressions ~r~ ty~V of Y with Qi(Y) = Y'WiY + B'ff + Y'Cf + Di is a family of indepen-~i~ ?Ji=l dent noncentral Wishart random operators are given. This version of Cochran's theorem combines certain results of Khatri (1980), Mathai and Provost (1992), Wong et al. (1991 }, and Wong andWang (1993) into a more general one. This result is further simplified to a verifiable version of Cochran's theorem for the case of nonnegative definite W~'s.