misc.tex

\chapter{Misc}
%\label{chap:misc}

% https://github.com/mitmath/matrixcalc

\section{Math}

\subsection{Derivative of a Quadratic Form}
\label{sec:derivative-quadratic-form}

The derivative of the quadratic form $x^\top A x$ is to be intended as the linear form $\nabla f(\bs{x})$ satisfying in the limit
\begin{equation*}
  f(\bs{x}+\delta\bs{x}) - f(\bs{x}) = \nabla f(\bs{x}) \delta x
\end{equation*}
of the scalar function $f:\mathbb{R}^n\rightarrow \mathbb{R}$.

When applied to the quadratic form $x^\top A x$, one obtains
\begin{equation}
  \label{eq:derivative-quadratic-form}
  \begin{aligned}
    & (x+\delta x)^\top A(x+\delta x) - x^\top Ax = \delta x^\top Ax + x^\top A\delta x \\
    &= x^\top A^\top \delta x + x^\top A\delta x = x^\top\left(A^\top + A\right)\delta x
  \end{aligned}
\end{equation}
where we have used the fact that the transpose of a real number coincides with the number.
Since this is a linear form in $v$, the vector is implicity intended and dropped and the result is usually given as $x^\top\left(A + A^\top\right)$.

% \section{Terminology}
% \label{sec:terminology}


% \subsubsection{Controllability and Observability/Detectability}
% \label{sec:controllability}

% Given the linear system
% \begin{equation*}
%   \begin{aligned}
%     x^+ = Ax + Bu \\
%     y = Cx + Du
%   \end{aligned}
% \end{equation*}
% the system is controllable if

% Controllability is usually given in terms of the matrix
% \begin{equation*}
%   \mathcal{C} =
%   \begin{bmatrix}
%     A & BA & B^2A & \ldots & B^{n-1}A
%   \end{bmatrix}
% \end{equation*}
% being full rank. Computing $\mathcal{C}$ rank is misleading and unfortunately, I feel it was not sufficiently strongly remarked in class:
% \begin{itemize}
% \item The rank is calculated by counting how many singular values of $\mathcal{C}$ are below a certain threshold. The threshold is usually set to be a multiple of the minimum resolution of a floating point number. Usually adding some noise to the matrix elements (or order $10^{-10}$ for Float64) makes the matrix full rank.
% \item Even computing the SVD 
% \end{itemize}

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