2017/10/24 by Rajeshwari Majumdar, Suman Majumdar, Majumdar, Rajeshwari +1 · 1 citation
Computer Science · Mathematics · #15A18 #46B28 #47A05 #60E99 #62B05 #62E15 #FOS: Mathematics #Matrix Theory and Algorithms #Statistics Theory (math.ST) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1710.09285
openalex publication_date 2017/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the orthogonal projection operator onto the range of the adjoint\nof a linear operator T can be represented as UT, where U is an invertible\nlinear operator. Using this representation we obtain a decomposition of a\nNormal random vector Y as the sum of a linear transformation of Y that is\nindependent of TY and an affine transformation of TY. We then use this\ndecomposition to prove that the conditional distribution of a Normal random\nvector Y given a linear transformation \TY is again a multivariate\nNormal distribution. This result is equivalent to the well-known result that\ngiven a k-dimensional component of a n-dimensional Normal random vector,\nwhere k<n, the conditional distribution of the remaining\n\(n-k\)-dimensional component is a \(n-k\)-dimensional\nmultivariate Normal distribution, and sets the stage for approximating the\nconditional distribution of Y given g\(Y\), where g is a\ncontinuously differentiable vector field.\n