MATH347DS L10: Some linear algebra applications

Overview

• Model reduction

• Stochastic matrices

• PageRank algorithm (initial Google search)

Model reduction

• Mass + spring systems lead to ODE system, $𝑴\ddot{𝒙}+𝑲𝒙=𝒇$, with $𝑴=\mathrm{diag}(m_1,\dots ,m_n)$, $𝑲\in {ℝ}^{n\times n}$, $𝒙\left(t\right):ℝ\to {ℝ}^n$ the trajectories of the point masses

• The point masses usually move in some correlated fashion, e.g., eigenmodes

• Construct a reduced model $𝒙=𝑩𝒚$, $𝑩\in {ℝ}^{m\times p}$, $p\ll n$, $𝑩$ orthonormal

• Take projection of system on $C\left(𝑩\right)$ (multiply by projection matrix $𝑷=𝑩{𝑩}^T$)

  ${\displaystyle \left({𝑩}^T𝑴𝑩\right)\ddot{𝒚}+({𝑩}^T𝑲𝑩)𝒚={𝑩}^T𝒇\iff 𝑵\ddot{𝒚}+𝑳𝒚=𝒈,𝑵,𝑳\in {ℝ}^{p\times p},𝒚,𝒈\in {ℝ}^p}$

Stochasdtic matrices

Definition. A Markov matrix $𝑹\in {ℝ}_{+}^{m\times m}$ (a.k.a. stochastic matrix) is a matrix with positive components and column vectors of unit 1-norm, ${||{𝒓}_j||}_1={\sum }_{i=1}^m{r}_{ij}=1$. The matrix entry $r_{ij}$ is the probability of transition from state $j$ to state $i$.

Definition. A Markov chain $\{{𝒑}_k\in {ℝ}^m,k=0,1,\dots \}$ (a.k.a. probability vectors) is a sequence of vectors generated by repeated application of a Markov matrix from the initial vector ${𝒑}_0$

Definition. The steady state of a Markov chain (if it exists) is the probability vector $𝒙$ for which

• Note: The steady state of a Markov chain is simply the eigenvector associated with eigenvalue $\lambda =1$

PageRank algorithm

• Given $m$ interlinked webpages, rank them in order of “importance”

• Let $x_i$ denote the importance score of page $i$, form vector $𝒙\in {ℝ}^m$

• Use the link structure of the web to determine “importance”

• ”Importance” (PageRank 1993):

  1. $x_i$ number of links to page $i$ (does not take into account importance of source)

  2. $x_i$ sum of importance scores of all pages linking to $i$

  3. $x_i$ sum of importance scores of pages that link to $i$ divided by the number of outgoing links on each page

     ${\displaystyle x_i=\sum _{j=1}^m{a}_{i,j}\frac{x_j}{n_j}=\sum _{j=1}^m{r}_{i,j}{x}_j\Rightarrow 𝒙=𝑹𝒙}$

     with $a_{i,j}=1$ if $j$ links to $i$, $a_{i,j}=0$ otherwise (adjacency matrix)

     ${\displaystyle r_{k,j}=\frac{1}{n_j}}$

• Brin-Page $𝑷=0.85𝑹+\frac{0.15}{m}\mathrm{ones}\left(m\right)$ (Stochastic matrices form a vector space!)