On the conditional distribution of a multivariate Normal given a transformation - the linear case.

Area of Science:

• Linear Algebra
• Probability Theory
• Statistical Inference

Background:

• Orthogonal projection operators are fundamental in linear algebra and functional analysis.
• Understanding conditional distributions of random vectors is crucial for statistical modeling and inference.

Purpose of the Study:

• To represent the orthogonal projection operator onto the range of the adjoint of a linear operator T.
• To derive a decomposition for normal random vectors that facilitates independent component analysis.
• To prove that the conditional distribution of a normal random vector given a linear transformation is also a multivariate normal distribution.

Main Methods:

• Utilizing the UT representation of the orthogonal projection operator.
• Decomposing a normal random vector Y into components independent of TY.
• Applying properties of multivariate normal distributions to derive conditional distributions.

Main Results:

• The orthogonal projection operator can be expressed as UT, where U is invertible.
• A linear operator was derived such that a component is independent of TY and an affine function of TY.
• The conditional distribution of a normal random vector Y given a linear transformation is proven to be multivariate normal.

Conclusions:

• The UT decomposition provides a new perspective on orthogonal projection operators.
• The established result on conditional distributions of normal random vectors has implications for statistical analysis.
• This work lays the foundation for approximating conditional distributions involving nonlinear vector fields.