\section{Establishing Physical Realizability}

In the absence of the $\bm{M}$, and $\bm{K}$ matrices, it is difficult to know what are the allowable negative values allowed for $\Delta \bm{M}$. In order to help approach this problem, it is necessary to (i) Establish a notion of physical realizability, and (ii) Develop bounds based on the prescribed definition of physical realizability. Therefore, this section works to provide an intuitive mathematical statement to capture the notion of physical realizability, based on system kinetic energy (KE). This idea is made clear in Definition \ref{def:KE}, and the mathematical implication of this definition is made clear in Remark \ref{lem:G}. 


\begin{defn}\label{def:KE} A linear mass-spring system that is claimed as being physically realizable requires at least, to possess positive KE.
\end{defn}

\begin{remark} \label{lem:G} An  $(\bm{M},\bm{K})$ system has positive KE if the eigenvalues of the $(\bm{I},\bm{M})$ system are positive. That is, if $w_i>0$ for all $i = 1,...,n$, then $\text{KE}\left[(\bm{M},\bm{K})\right]>0$.
\end{remark}

\begin{proof}
As shown in Equation \ref{eqn:mass_system}, the eigenpairs for the mass-only system: $(\bm{I},\bm{M})$, are $(w_i, \bm{\phi}_i)$, with eigenvalues: $w_i$, and eigenvector: $\bm{\phi}_i$. Consider the following diagonalization:

\begin{align*} 
\bm{M} = \bm{\Phi}^{-1} \bm{D \Phi}, 
\end{align*} 

where $\bm{\Phi}=[\bm{\phi}_1, \ldots, \bm{\phi}_n]$ is the matrix of eigenvector columns and $\bm{D} = \text{diag}(w_1,\ldots,w_n)$.  Since $\braces{\bm{\phi}_i}_{i=1}^n$ forms a basis over $\mathbb{R}^n$, any $\dot{\bm{x}} \in \mathbb{R}^n$ velocity vector can be expressed as $\dot{\bm{x}} = \sum_{i=1}^n a_i \bm{\phi}_i$, where $a_i\in\mathbb{R}$ are constants. Now, $\forall \dot{\bm{x}} \in \mathbb{R}^N$, $\text{KE}\left[(\bm{M},\bm{K})\right]$, w.r.t. $\dot{\bm{x}}$ can be defined as follows:  

\begin{align} \text{KE}\left[(\bm{M},\bm{K});\dot{\bm{x}}\right] &\coloneqq \frac{1}{2} \ip{\dot{\bm{x}}}{\bm{M}\dot{\bm{x}}} \nonumber\\ &= \ip{\sum_{i=1}^N a_i \bm{\phi}_i}{\sum_{j=1}^N a_j \bm{M \phi}_j}\nonumber\\ &\stackrel{(i)}{=} \sum_{i=1}^N \sum_{j=1}^N a_i a_j \ip{\bm{\phi}_i}{ w\bm{\phi}_j} \\ &\stackrel{(ii)}{=}\sum_{i=1}^N w_i a_i^2 \nonumber\\  & > 0, \text{ if $w_i > 0$.} \nonumber 
\end{align} 

where (i) follows from linearity of the inner product, and (ii) follows from the orthogonality property of eigenvectors.
\end{proof}

\textbf{Summary: } \textit{What Remark \ref{lem:G} says, is that if the eigenvalues of $\bm{M}$ are positive, then the corresponding $(\bm{M},\bm{K})$ system is physically realizable through a KE argument. In other words, if $\bm{M}$ is positive definite (denoted: $\bm{M} \succ 0$) then the system is physically realizable. We will use this idea to guide our lower bounds.}
{"config": "arxiv", "file": "2006.16802/Establishing_Physical_Realizability.tex"}