# Shae's Ramblings

Stuff I find slightly meaningful and close to achievable

## Multivariate gaussian and matrix-vector products: basics

Assume you have a multivariate gaussian vector $\mathbf{x}$, so that $p(\mathbf{x}) = \mathcal{N}(\pmb{\mu}, \pmb{\Sigma})$. Suppose we have a fixed matrix $\mathbf{A}$ and vector $\mathbf{c}$.
What is the distribution of

• The matrix-vector product $\mathbf{A}\mathbf{x}$ ?
• The inner product $\mathbf{c}^\mathsf{T}\mathbf{x}$ ?
• The outer products $\mathbf{x}\mathbf{c}^\mathsf{T}, \mathbf{c}\mathbf{x}^\mathsf{T}$ ?

## Matrix-vector product

Because $\pmb{\Sigma}$ is positive-semi-definite, it has a cholesky factor $\mathbf{L} = \mathrm{chol}(\pmb{\Sigma})$, and further the equality

$$\mathbf{x} = \pmb{\mu} + \mathbf{L}\mathbf{z}\,,\qquad \mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$$

holds. It is then easy to compute the matrix product and derive its distribution:

$$\mathbf{A}\mathbf{x} = \mathbf{A}\pmb{\mu} + \mathbf{A}\mathbf{L}\mathbf{z}\,,\qquad \mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$$

So that applying the same trick in reverse yields that $\mathbf{A}\mathbf{x} \sim \mathcal{N}(\tilde{\pmb{\mu}}, \tilde{\pmb{\Sigma}})$ with

$$\tilde{\pmb{\mu}} = \mathbf{A}\pmb{\mu}\,,\qquad \tilde{\pmb{\Sigma}} = \mathbf{A}\mathbf{L}\mathbf{L}^\mathsf{T}\mathbf{A}^\mathsf{T} = \mathbf{A}\pmb{\Sigma}\mathbf{A}^\mathsf{T}$$

## Inner product

A consequence of this first result is obtained by taking $\mathbf{A} = \mathbf{c}^\mathsf{T} \in \mathbb{R}^{1 \times p}$
Thus, we have the result

$$\mathbf{c}^\mathsf{T} \mathbf{x}\;\sim\;\mathcal{N}\left( \mathbf{c}^\mathsf{T}\pmb{\mu}\,,\;\mathbf{c}^\mathsf{T}\pmb{\Sigma}\mathbf{c}\right)$$

## Outer product

$$\mathbf{x}\mathbf{c}^\mathsf{T} =\pmb{\mu}\mathbf{c}^\mathsf{T} + \mathbf{L}\mathbf{z}\mathbf{c}^\mathsf{T}\,,\qquad \mathbf{z} \sim \mathcal{N}(\mathbf{0}, \mathbf{I})$$