UnitBlockAngular.jl

Specialized linear algebra for unit block-angular matrices. It is primarily intended for use within Tulip.jl

Overview

Uni block-block angular matrices have the form

$$ A = \begin{bmatrix} B_{0} & B_{1} & B_{2} & \dots & B_{R}\\ 0 & e^{T} \\ 0 & & e^{T}\\ \vdots & & & \ddots\\ 0 & & & & e^{T} \end{bmatrix} $$

This package exports the UnitBlobkAngularMatrix type, and extends the 5-arg mul!.

Matrix-vector multiplication

$y = A \times x$ writes

$$ \large \begin{bmatrix} y_{0}\ y_{1}\ y_{2} \ \vdots\ y_{R} \end{bmatrix}

\begin{bmatrix} B_{0} & B_{1} & B_{2} & \dots & B_{R}\ 0 & e^{T} \ 0 & & e^{T}\ \vdots & & & \ddots\ 0 & & & & e^{T} \end{bmatrix} \begin{bmatrix} x_{0}\ x_{1}\ x_{2} \ \vdots\ x_{R} \end{bmatrix}

\begin{bmatrix} \displaystyle \sum_{i=0}^{R}B_{i}x_{i}\ e^{T}x_{1}\ e^{T}x_{2}\ \vdots\ e^{T}x_{R} \end{bmatrix}

\begin{bmatrix} B_{0} x_{0} + B x_{-0}\ e^{T}x_{1}\ e^{T}x_{2}\ \vdots\ e^{T}x_{R} \end{bmatrix} $$

$x = A^{T} \times y$ writes

$$ \Large \begin{bmatrix} x_{0}\ x_{1}\ x_{2} \ \vdots\ x_{R} \end{bmatrix}

\begin{bmatrix} B_{0}^{T} & 0 &0 & \dots & 0\ B_{1}^{T} & e\ B_{2}^{T} &&e\ \vdots &&& \ddots\ B_{R}^{T} & &&&e \end{bmatrix} \begin{bmatrix} y_{0}\ y_{1}\ y_{2} \ \vdots\ y_{R} \end{bmatrix}

\begin{bmatrix} B_{0}^{T}y_{0}\ B_{1}^{T}y_{0} + y_{1} e\ B_{2}^{T}y_{0} + y_{2} e\ \vdots\ B_{R}^{T}y_{0} + y_{R} e\ \end{bmatrix}

\begin{bmatrix} B_{0}^{T}y_{0}\ B^{T} y_{-0} \end{bmatrix} + \begin{bmatrix} 0\ y_{1} e\ y_{2} e\ \vdots \ y_{R} e\ \end{bmatrix} $$

Factorization

The normal equations write

$$ A \times \Theta \times A^{T} = \begin{bmatrix} \sum_{i=0}^{R} B_{i} \Theta_{i} B_{i}^{T} & B_{1} \theta_{1} & B_{2}\theta_{2} & \dots & B_{R} \theta_{r}\\ \theta_{1}^{T} B_{1}^{T} & e^{T}\theta_{1} \\ \theta_{2}^{T} B_{2}^{T} & & e^{T}\theta_{2}\\ \vdots & & & \ddots\\ \theta_{R}^{T} B_{R}^{T} & & & & e^{T}\theta_{R} \end{bmatrix} $$