mathematical_background.tex

\chapter{Mathematical Preliminaries}
\label{cha:math-prel}
\section{Graph Laplacians}
\label{sec:graph-laplacians}
We now introduce the concept of the Laplacian matrix of a graph. Our
exposition will be very superficial. For a comprehensive account
of graph Laplacians, see
\citep{chung05:_laplac_cheeg,cvetkovic80:_spect_graph_theor_applic}. \\
\\
%
\noindent Let $G = (V,E,\omega)$ be a simple, undirected graph with
vertex set $V$, edge set $E$ and similarity measure $\omega \colon E \mapsto
\mathbb{R}^{\geq 0}$. If $u$ and $v$ are vertices of $G$, we write $u
\sim v$ whenever $\{u,v\} \in E$. The degree of a vertex $v$ is
defined as $\deg(v) = \sum_{u \sim v}{\omega(\{u,v\})}$ and the volume
of $G$ is $\mathrm{Vol}(G) = \sum_{v \in V}{\deg(v)}$.  We denote by
$N$ the number of vertices of $G$. We define $D = (d_{ij})$ as the $N
\times N$ diagonal matrix with diagonal entries $d_{vv} = \deg(v)$.
\begin{definition}
  \label{def:1}
  Let $G = (V,E,\omega)$ be a simple, undirected graph with similarity
  measure $\omega$. The {\em combinatorial} Laplacian of $G$ is the
  matrix $L = L(G)$ with entries
  \begin{equation}
    \label{eq:1}
    L_{uv} = \begin{cases}
      - \omega(\{u,v\}) & \text{if $u \not = v$ and $u \sim v$} \\
      \deg(u) & \text{if $u = v$} \\
      0 & \text{otherwise}
    \end{cases}
  \end{equation}
  The {\em normalized} Laplacian of $G$ is the matrix $\mathcal{L} =
  \mathcal{L}(G)$ with entries
  \begin{equation}
    \label{eq:2}
    \mathcal{L}_{uv} = \begin{cases}
      - \tfrac{\omega(\{u,v\})}{\sqrt{\deg(u)}\sqrt{\deg(v)}} & \text{if $u \not = v$ and $u \sim v$} \\
      1 & \text{if $u = v$} \\
      0 & \text{otherwise}
    \end{cases}
  \end{equation}
\end{definition}
The following proposition lists some simple properties of the
combinatorial and normalized Laplacians. 
\begin{proposition}
  \label{prop:1}
  Let $G = (V,E,\omega)$ be a simple, undirected graph and $L$ and
  $\mathcal{L}$ be its combinatorial and normalized Laplacians,
  respectively. We have
  \begin{itemize}
  \item $L$ and $\mathcal{L}$ are symmetric, positive
    semi-definite matrices.
  \item $\mathcal{L} = D^{-1/2} L D^{-1/2}$
  \item The number of connected components of $G$ is equal to the
    number of zero eigenvalues of either $L$ or $\mathcal{L}$.
  \item The eigenvalues of $\mathcal{L}$ are at most $2$. 
  \end{itemize}
\end{proposition}
\section{Finite Markov chains}
\label{sec:finite-markov-chain}
\begin{definition}
  \label{def:6}
  Let $\Omega$ be a finite or countably infinite set and
  $\mathbb{Z}^{*}$ be the set of non-negative integers. A sequence
  $\mathbf{X} = (X_n)_{n \in \mathbb{Z}^{*}}$ of random variables with values in
  $\Omega$ is a {\em Markov chain} if
  \begin{equation}
    \label{eq:8}
    \mathbb{P}[X_{n+1} = j \, | \, X_n = i, X_{n-1} = i_{n-1},
    \dots, X_0 = i_0] = \mathbb{P}[X_{n+1} = j \, | \, X_n = i] =
    p_{ij}
  \end{equation}
  for all $n \geq 0$ and all states $i_0, i_1, \dots, i_{n-1}, i,
  j$. The matrix $\mathbf{P}$, possibly infinite, with entries
  $\mathbf{P}(i,j) = p_{ij}$ is then termed the transition matrix of
  $(X_n)_{n \in \mathbb{Z}^*}$.
\end{definition}
Let $\mathbf{X} = (X_n)_{n \in \mathbb{Z}^*}$ be a Markov
chain. Denote by $p_{ij}^{(n)}$ the probability of going from state
$i$ to state $j$ in $n$ steps, i.e.,
\begin{equation}
  \label{eq:11}
  p_{ij}^{(n)} = \mathbb{P}[ X_{n + m } = j \, | \, X_m = i]
\end{equation}
for all $i, j \in \Omega$, and $m,n \in \mathbb{Z}^{*}$. Then
$p_{ij}^{(n)}$ satisfy the {\em Chapman-Kolmogorov equation}
\begin{equation}
  \label{eq:12}
  p_{ij}^{(m+n)} = \sum_{k \in \Omega}{p_{ik}^{(m)}p_{kj}^{(n)}}
\end{equation}
for all $m,n \in \mathbb{Z}^{*}$. Thus if $\mathbf{P}^{(n)}$ is the
matrix with entries $p_{ij}^{(n)}$, then $\mathbf{P}^{(m+n)} =
\mathbf{P}^{(m)}\mathbf{P}^{(n)}$. Because $\mathbf{P}^{(1)} =
\mathbf{P}$, we have
\begin{equation}
  \label{eq:13}
  \mathbf{P}^{(n)} = \mathbf{P}^{n}
\end{equation}
The behavior of a Markov chain $\mathbf{X} = (X_n)_{n \in
  \mathbb{Z}^{*}}$ is thus completely specified by its transition
matrix $\mathbf{P}$. We can therefore view a Markov chain as being a
sequence of random variables generated by a transition matrix
$\mathbf{P}$. This view will be most helpful in the context of this
work. However, because the transition matrix $\mathbf{P}$ only describes
the conditional probabilities, in order for us to compute the marginal
probabilities $\mathbb{P}[X_n = j]$, we need to specify an initial
distribution for $X_0$.

\begin{definition}
  \label{def:5}
  Let $\mathbf{X}$ be a Markov chain with state space
  $\Omega$. The initial distribution $\mu$ of $\mathbf{X}$ is a probability
  distribution on $\Omega$ such that 
  \begin{equation}
    \label{eq:14}
    \mu(i) = \mathbb{P}[X_0 = i]
  \end{equation}
  for all $i \in \Omega$. 
\end{definition}

\begin{definition}
  \label{def:7}
  Let $\mathbf{X}$ be a Markov chain with state space $\Omega$. Let
  $i$ and $j$ be elements of $\Omega$. $j$ is
  {\em accessible} from $i$, denoted as $i \rightarrow j$, if there
  exists a $n \in \mathbb{Z}^{*}$ such that $p_{ij}^{(n)} > 0$. If $i
  \rightarrow j$ and $j \rightarrow i$, then we say that $i$ and $j$
  {\em communicate}, and we write $i \leftrightarrow j$. A Markov chain is
  {\em irreducible} if $i \leftrightarrow j$ for any $i,j \in \Omega$.
\end{definition}
\begin{definition}
  \label{def:2}
  The stationary distribution $\pi$ of
  $\mathbf{X}$, if it exists, is a probability distribution on
  $\Omega$ such that
  \begin{equation}
    \label{eq:15}
    \pi(j) = \sum_{i \in \Omega}{\pi(i) p_{ij}}
  \end{equation}
  for any $j \in \Omega$. 
\end{definition}

\begin{proposition}
  \label{prop:3}
  If $\mathbf{X}$ is an irreducible Markov chain with state space
  $\Omega$, then there exists a unique stationary distribution $\pi$
  of $\mathbf{X}$, and that $\pi(i) > 0$ for all $i \in \Omega$. 
\end{proposition}

\begin{definition}
  \label{def:3}
  Let $\mathbf{X}$ be a Markov chain
  with transition matrix $\mathbf{P}$. Define 
  \begin{equation}
    \label{eq:5}
    \tau_i = \min\{ t \geq 0 \colon X_t = i \}, \qquad \tau_i^{+} \min
    \{ t \geq 1 \colon X_t = i \}.
  \end{equation}
  The expected first passage time from $i$ to $j$, denoted by
  $\mathbb{E}_{i}[\tau_j]$, is defined as
  \begin{equation}
    \label{eq:6}
    \mathbb{E}_{i}[\tau_j] = \sum_{t = 0}^{\infty}{t \, \mathbb{P}(\tau_j =
      t \,|\, X_0 = i)}.
  \end{equation}
  The expected first return time from $i$ to $i$, denoted by
  $\mathbb{E}_{i}[\tau_i^{+}]$, is defined as
  \begin{equation}
    \label{eq:7}
    \mathbb{E}_{i}[\tau_i^{+}] = \sum_{t = 1}^{\infty}{t \,
      \mathbb{P}(\tau_v^{+} = t \,|\, X_0 = i)}.
  \end{equation}
  $\tau_i$ and $\tau_{i}^{+}$ as declared above are examples of {\em
    stopping times}. 
\end{definition}

\begin{proposition}
  \label{prop:2}
  Let $\mathbf{X}$ be an irreducible
  Markov chain with transition matrix $\mathbf{P}$ and stationary
  distribution $\pi$. We then have that
  \begin{equation}
    \label{eq:9}
    \mathbb{E}_{i}[\tau_i^{+}] = \frac{1}{\pi(i)}.
  \end{equation}
\end{proposition}

\begin{definition}
  \label{def:9}
  Let $\mathbf{X}$ be an irreducible Markov chain with transition
  matrix $\mathbf{P}$ and stationary distribution
  $\pi$. $\hat{\mathbf{P}} = (\hat{p}_{ij})$ is said to be the {\em
    time reversal} of $\mathbf{P}$ if, for all pairs $i,j \in \Omega$,
  one has
  \begin{gather}
    \label{eq:16}
    \pi(i) p_{ij} = \pi(j) \hat{p}_{ji}, \\
    \intertext{or, in other words}
    \label{eq:78}
    \hat{\mathbf{P}} = \bm{\Pi}^{-1} \mathbf{P}^{T} \bm{\Pi}. 
  \end{gather}
  $\mathbf{P}$ is said to be {\em time-reversible} if
  $\hat{\mathbf{P}} = \mathbf{P}$.
\end{definition}
Now $\hat{\mathbf{P}}$ also defines a Markov chain
$\hat{\mathbf{X}}$. $\hat{\mathbf{X}}$ will be termed the
time-reversed Markov chain with respect to $\mathbf{X}$. $\pi$ is also
the stationary distribution of $\hat{\mathbf{P}}$ and that
\begin{equation}
  \label{eq:17}
  \mathbb{P}[X_n = j, \dots, X_0 = i] = \mathbb{P}[\hat{X}_0 = i,
  \dots, \hat{X}_n = j] 
\end{equation}
where the initial distribution of $X_0$ and $\hat{X}_0$ are both
identical to the stationary distribution $\pi$.
\section{Random walks on graphs}
\label{sec:random-walks-graphs}
Let $G = (V,E,\omega)$ be a simple, undirected graph. We define the transition
matrix $\mathbf{P}_G = (p_{uv})$ of a Markov chain with state space $V$ as follows
\begin{equation}
  \label{eq:20}
  p_{uv} = \begin{cases}
    \tfrac{\omega(\{u,v\})}{\deg(u)} & \text{if $u \sim v$} \\
    0 & \text{otherwise}
  \end{cases}
\end{equation}
We now note some properties of the Markov chain $\mathbf{X}$ generated
by $\mathbf{P}_G$.
\begin{proposition}
  \label{prop:15}
  Let $G$ be an undirected graph and $\mathbf{P}$ be the transition
  matrix on $G$. Let $\mathbf{X}$ be the Markov chain generated by
  $\mathbf{P}$. Then 
\begin{itemize}
\item $\mathbf{X}$ is irreducible if and only if $G$ is connected.
\item If $\mathbf{X}$ is irreducible, $\pi(v) =
  \tfrac{\deg(v)}{\mathrm{Vol}(G)}$ for all $v \in V$.
\item $\mathbf{P}$ is time-reversible. Therefore, $\bm{\Pi}\mathbf{P} = \mathbf{P}^{T}\bm{\Pi}$.
\end{itemize}
\end{proposition}
%
We can also define the transition matrix $\mathbf{P}_G$ when $G$ is
directed. $\mathbf{P}_G$ will have entries
\begin{equation}
  \label{eq:18}
  p_{uv} = \begin{cases}
    \tfrac{\omega(e)}{\deg(u)} & \text{if $e = (u,v) \in E$} \\
    0 & \text{otherwise}
  \end{cases}
\end{equation}
If $G$ is directed, then $\mathbf{X}$ is irreducible if and only if
$G$ is strongly connected. However, $\mathbf{P}$ is in general not
time reversible and there is no explicit expression for the
stationary distribution $\pi$ of $\mathbf{P}$. \\ \\
%
%
\noindent Let $G = (V,E)$ be a graph, directed or undirected, and $\mathbf{P}$
be its transition matrix. A function $f \colon V \mapsto \mathbb{R}$
is {\em harmonic} at $v \in V$ if
\begin{equation}
  \label{eq:10}
  f(v) = \sum_{w \in V}{\mathbf{P}(v,w) f(w)}
\end{equation}
$f$ is harmonic on $V$ if it is harmonic for all $v \in V$. If $\mathbf{P}$
is irreducible, we have a simple characterization for harmonic
functions on $V$. Specifically,
\begin{lemma}
  \label{lem:1}
  Suppose that $\mathbf{P}$ is irreducible. A function $f \colon V \mapsto
  \mathbb{R}$ is harmonic on $V$ if and only if $f$ is constant on
  $V$. 
\end{lemma}
\begin{proof}
  It's easy to see that if $f$ is constant on $V$ then it is also
  harmonic on $V$. Thus, let us assume that $f$ is harmonic on $V$.
  Let $v_*$ be a node such that $f(v_*) \geq f(w)$ for all $w \in
  V$. Because $f$ is harmonic, from Eq.~\eqref{eq:10} we have that $f(w)
  = f(v_*)$ for all $w$ such that $\mathbf{P}(v_*,w) > 0$. We thus see
  that every vertex $w$ that is accessible from $v_*$ will satisfy
  $f(w) = f(v_*)$. Because $\mathbf{P}$ is irreducible, $f(v_*) = f(w)$ for
  all $w \in V$. $f$ is thus constant on $V$.
\end{proof}

\begin{proposition}
  \label{prop:6}
  Let $G = (V,E)$ be a graph and $\mathbf{P}$ be its transition
  matrix. Suppose that the Markov chain defined by $\mathbf{P}$ is
  regular. Then there exists a unique stationary distribution $\pi$ of
  $\mathbf{P}$. Furthermore, if $\mathbf{Q} = \mathbf{1} \mathbf{\pi}^{T}$ is the
  matrix with each row being the stationary distribution, then
  \begin{equation}
    \label{eq:22}
    \lim_{k \rightarrow \infty}(\mathbf{P} - \mathbf{Q})^{k} = 0 
  \end{equation}
  Eq.~\eqref{eq:22} is equivalent to $\rho(\mathbf{P}-\mathbf{Q}) < 1$
  where $\rho(\mathbf{P}-\mathbf{Q})$ is the spectral radius of
  $\mathbf{P} - \mathbf{Q}$.
\end{proposition}

\begin{proposition}
  \label{prop:7}
  Let $G = (V,E)$ be a graph and $\mathbf{P}$ be its transition
  matrix. Suppose that $\mathbf{P}$ is regular. Then the matrix $\mathbf{Z} =
  (\mathbf{I} - \mathbf{P} + \mathbf{Q})^{-1}$ exists and is given by
  \begin{equation}
    \label{eq:28}
    \mathbf{Z} = \sum_{k=0}^{\infty}(\mathbf{P} - \mathbf{Q})^{k} = \mathbf{I} +
    \sum_{k=1}^{\infty}(\mathbf{P}^{k} - \mathbf{Q})
  \end{equation}
  
\end{proposition}
\begin{proof}
  Because $\mathbf{P}$ is regular, by Proposition \ref{prop:6}, $\lim_{k
    \rightarrow \infty}(\mathbf{P} - \mathbf{Q})^{k} = 0$. Thus, $\mathbf{Z}$ has
  an expansion in terms of a Neumann series
  \begin{equation}
    \label{eq:29}
    \mathbf{Z} = \sum_{k=0}^{\infty}(\mathbf{P} - \mathbf{Q})^{k}
  \end{equation}
  Because $\mathbf{P}\mathbf{Q} = \mathbf{P}\bm{1}^{T}\bm{\pi} =
  \bm{1}^{T}\bm{\pi} = 
  \bm{1}^{T}\bm{\pi}\mathbf{P} = \mathbf{Q}\mathbf{P} = \mathbf{Q}$,  
  one has $(\mathbf{P} - \mathbf{Q})^{k} = \mathbf{P}^{k} - \mathbf{Q}$ for $k \geq
  1$. Eq.~\eqref{eq:28} thus follows. 
\end{proof}
The matrix $\mathbf{Z}$ is termed the {\em fundamental matrix}
\citep{kemeny83:_finit_markov_chain}. Some properties of
$\mathbf{Z}$ are given in the following proposition.
\begin{proposition}
  \label{prop:8}
  Let $\mathbf{P}$ be the transition matrix of a regular Markov chain and
  $\mathbf{Z}$ be its fundamental matrix. We have
  \begin{enumerate}[(i)]
  \item $\mathbf{P}\mathbf{Z} = \mathbf{Z} - \mathbf{I} + \mathbf{Q}$. 
  \item $(\mathbf{I} - \mathbf{P})\mathbf{Z} = \mathbf{I} - \mathbf{Q}$.
  \item $\mathbf{Z} \mathbf{J} = \mathbf{J}$. 
  \end{enumerate}
\end{proposition}
\begin{proof}
  $\mathbf{P}\mathbf{Z} = \mathbf{P} - \sum_{k=1}^{\infty}(\mathbf{P}^{k+1} - \mathbf{Q})
  = \mathbf{Z} - \mathbf{I} + \mathbf{Q}$. (i) and (ii) thus follow. For (iii),
  note that $\mathbf{P}^{k}\mathbf{J} = \mathbf{Q}\mathbf{J} = \mathbf{J}$. 
\end{proof}
\section{Distance geometry}
\label{sec:distance-geometry}
We discuss in this section some notation and results regarding
distance matrices. The notion of a Euclidean distance matrix (EDM) is
of particular importance to our discussion and is defined in
Definition \ref{def:1}. We then introduce two linear transformations
between matrices, the $\kappa$ transform and the $\tau$ transform.
Schoenberg's characterization
\citep{schoenberg35:_remar_mauric_frech_artic_sur} of Euclidean
distance matrices in terms of positive semidefinite matrices is
stated in Theorem \ref{thm:5}. We also state some simple results
regarding the $\kappa$ transforms that are useful in the context of
this work.

\begin{definition}
  \label{def:10}
  Let $\Delta = (\delta_{ij}) \in M_n(\mathbb{R})$. $\Delta$ is a Type 1
  Euclidean distance matrix (EDM-$1$) if and only if there exists a
  positive integer $p$ and $x_1, x_2, \dots, x_n \in \mathbb{R}^{p}$
  such that $\delta_{ij} = \| x_i - x_j \|$. $\Delta$ is a Type 2
  Euclidean distance matrix (EDM-$2$) if and only if there exists a
  $p$ and $x_1, x_2, \dots, x_n \in \mathbb{R}^{p}$ such that
  $\delta_{ij} = \|x_i - x_j\|^{2}$. The {\em embedding dimension} of
  $\Delta$ is the minimum $p$ such that a configuration of points
  $x_1, x_2, \dots, x_n$ exists with the desired property.
\end{definition}

\begin{definition}
  \label{def:11}
  Let $\mathbf{A} \in M_n(\mathbb{R})$. Define a linear mapping $\tau \colon M_n(\mathbb{R})
  \mapsto M_n(\mathbb{R})$ by
  \begin{equation}
    \label{eq:83}
    \tau(\mathbf{A}) = - \frac{1}{2} \Bigl(\mathbf{I} -
    \frac{\mathbf{J}}{n}\Bigr)\mathbf{A} \Bigl(\mathbf{I} - \frac{\mathbf{J}}{n}\Bigr)
  \end{equation}
  If $a_{ij}$ are the entries of $\mathbf{A}$ then
  \begin{equation}
    \label{eq:56}
    b_{ij} = -\frac{1}{2}\Bigl(a_{ij} - \frac{1}{n}\sum_{j=1}^{n}a_{ij} -
    \frac{1}{n}\sum_{i=1}^{n}{a_{ij}} +
      \frac{1}{n^2}\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}\Bigr)
  \end{equation}
  are the entries of $\mathbf{B} = \tau(\mathbf{A})$. $\tau$
  is a continuous mapping from $M_n(\mathbb{R})$ to
  $M_n(\mathbb{R})$. 
\end{definition}
%
The following result provides a necessary and sufficient condition for
$\Delta$ to be an EDM-2 matrix.
\begin{theorem}[\citet{schoenberg38:_metric,young38:_discus}]
  \label{thm:5}
  $\Delta$ is an EDM-2 with embedding dimension $p$ if and only
  if $\mathbf{B} = \tau(\Delta)$ is positive semidefinite with rank
  $p$. 
\end{theorem}

\begin{definition}
  \label{def:12}
  Let $\mathbf{A} \in M_n(\mathbb{R})$. Define a linear mapping
  $\kappa \colon M_n(\mathbb{R}) \mapsto M_n(\mathbb{R})$ by
  \begin{equation}
    \label{eq:61}
    \kappa(\mathbf{A}) = \mathbf{J}\mathbf{A}_{\mathrm{dg}} -
    \mathbf{A} - \mathbf{A}^{T} + \mathbf{A}_{\mathrm{dg}}\mathbf{J}
  \end{equation}
  where $\mathbf{A}_{\mathrm{dg}}$ is the diagonal matrix obtained by
  setting the off-diagonal entries of $\mathbf{A}$ to $0$. If $a_{ij}$
  are the entries of $\mathbf{A}$ then
  \begin{equation}
    \label{eq:70}
    b_{ij} = a_{ii} - a_{ij} - a_{ji} + a_{jj}
  \end{equation}
  are the entries of $\mathbf{B} = \kappa(\mathbf{A})$. $\kappa$ is
  also a continuous mapping from $M_n(\mathbb{R})$ to
  $M_n(\mathbb{R})$. 
\end{definition}

\begin{proposition}
  \label{prop:16}
  The $\kappa$ transform has the following properties.
  \begin{enumerate}[i]
  \item Let $\mathcal{C} = \{ \mathbf{A} \in S_n(\mathbb{R}) \colon
    \mathbf{C}\bm{1}_{n}^{T} = \bm{0} \}$ be the set of symmetric
    matrices with zero row sums and let $\mathcal{D} = \{ \Delta \in
    S_n(\mathbb{R}) \colon \Delta_{\mathrm{dg}} = 0 \}$ be the set of
    symmetric hollow matrices. Then $\kappa$ and $\tau$ are inverse
    mappings between $\mathcal{C}$ and $\mathcal{D}$, i.e.,
  \begin{gather}
    \label{eq:55}
    \mathbf{A} \in \mathcal{C}
    \Longrightarrow \Delta = \kappa(\mathbf{A}) \in \mathcal{D}, \,\,
    \mathbf{A} = \tau(\Delta) \\
    \Delta \in \mathcal{D} \Longrightarrow \mathbf{A} = \tau(\Delta)
    \in \mathcal{C}, \,\, \Delta = \kappa(\mathbf{A})
  \end{gather}
  \item $\kappa(\mathbf{J}) = 0$. More generally,
    $\kappa(\bm{a}\bm{1}^{T}) = \kappa(\bm{1}\bm{b}^{T}) = 0$
    for any vector $\bm{a}$, $\bm{b}$.
  \item Let $\tilde{\mathbf{X}}$ be the double centering of
    $\mathbf{X}$, i.e.,
  \begin{equation}
    \label{eq:71}
    \tilde{\mathbf{X}} = \Bigl(\mathbf{I} - \frac{\mathbf{J}}{n}\Bigr)\mathbf{X} \Bigl(\mathbf{I} - \frac{\mathbf{J}}{n}\Bigr)
  \end{equation}
  Then $\kappa(\tilde{\mathbf{X}}) = \kappa(\mathbf{X})$.
  \end{enumerate}
\end{proposition}
Part (i) of Proposition \ref{prop:16} is from
\citep{critchley88:_certain_linear_mappin}. Part (ii) and (iii)
follow directly from the definition of the $\kappa$ transform. 
\begin{proposition}
  \label{prop:18}
  Let $\mathbf{A} \in S_n(\mathbb{R})$ be a positive semidefinite
  matrix. Then $\Delta = \kappa(\mathbf{A})$ is EDM-2.
\end{proposition}
\begin{proof}
  The double centering $\tilde{\mathbf{A}}$ of $\mathbf{A}$ is a
  matrix in $\mathcal{C}$. By Proposition
  \ref{prop:16}, $\Delta = \kappa(\mathbf{A}) =
  \kappa(\tilde{\mathbf{A}})$ and $\tilde{\mathbf{A}} =
  \tau(\Delta)$. Now $\mathbf{A} \succeq 0$ implies
  $\tilde{\mathbf{A}} \succeq 0$. By Schoenberg's criterion, $\Delta =
  \kappa(\tilde{\mathbf{A}})$ is EDM-2. 
\end{proof}
%
\section{Matrix analysis}
\label{sec:matrix-analysis}
We list here some results in matrix analysis that are useful within
the scope of this work.
%
Let $\mathbf{A} = (a_{ij})$ be an $n \times n$ matrix with real
entries $a_{ij}$. Denote by $R_i$ the sum $\sum_{j \not = i}{|a_{ij}|}$ of
off-diagonal elements in row $i$. 
%
\begin{theorem}[\cite{gersgorin31:_uber_abgren_eigen_matrix}]
  \label{thm:1}
  Let $\mathbf{A}$ be an $n \times n$ matrix with off-diagonal row
  sums $R_i$. Then the eigenvalues of $\mathbf{A}$ lie in the set
  \begin{equation}
    \label{eq:23}
    \bigcup \{z \in \mathbb{C} \colon |z - a_{ii}| \leq R_i \}
  \end{equation}
\end{theorem}
\begin{definition}
  \label{def:4}
  The matrix $\mathbf{A}$ is said to be diagonally dominant if
  $|a_{ii}| \geq R_i$ for all $i$ and strictly diagonally dominant if
  $|a_{ii}| > R_i$ for all $i$.
\end{definition}
If $\mathbf{A}$ is diagonally dominant, then by Ger\u{s}gorin's
circle theorem, the eigenvalues of $\mathbf{A}$ have nonnegative real
parts. If $\mathbf{A}$ is strictly diagonally dominant, then the
eigenvalues of $\mathbf{A}$ has positive real parts. 
%
\begin{definition}
  \label{def:8}
  Let $Z_n \subset M_{n}(\mathbb{R})$ be the set of matrices with
  non-positive off-diagonal entries, i.e.,
  \begin{equation}
    \label{eq:24}
    Z_n = \{ \mathbf{A} = (a_{ij}) \in M_{n}(\mathbb{R}) \colon a_{ij}
    \leq 0 \,\, \text{if $i \not = j$} \}
  \end{equation}
 A matrix $\mathbf{A} \in Z_n$ is called an $M$-matrix if $A$ is
 positive stable, i.e., if the eigenvalues of $\mathbf{A}$ has
 positive real parts.
\end{definition}
A relationship between $M$-matrices and non-negative matrices is given
by the following result \citep[\S 2.5]{horn94:_topic_in_matrix_analy}.
\begin{theorem}
  \label{thm:2}
  $\mathbf{A} \in Z_n$ is an $M$-matrix if and only if $\mathbf{A}$ is
  not singular and $\mathbf{A}^{-1} \geq 0$.  
\end{theorem}
%%% Local Variables: 
%%% mode: latex
%%% TeX-master: "dissertation"
%%% End: