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Mathematics-Online course: Linear Algebra - Normal Forms - Diagonalisation

Powers of Matrices

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Let $ A$ be a matrix having an eigenvalue $ \lambda$ of largest absolute value with associated eigenvector $ u$. If vector $ x$ has a nontrivial component in the eigenspace of $ \lambda$, that is, if $ x=cu+v$ with $ c\ne0$ and $ u\nparallel v$, then we have

$\displaystyle A^n x = \lambda^n (c u + o(1)),\quad n\to\infty \,. $

(Authors: App/Burkhardt/Höllig)
The figure shows the annual change of market shares $ x_i$ of competing enterprises. Enterprise A, for example, gains $ 80\%$ annually of the market shares of enterprise D, and enterprise C, opening up additional sales potentials, increases its market shares by $ 90\%$ but, at the same time, loses market shares to competitors A and D.

\includegraphics[width=.6\moimagesize]{b_marktanteile}

From the figure we find the new market shares:

$\displaystyle A_$new $\displaystyle = 0.7 A + 0.4 C + 0.8 D$    
$\displaystyle B_$new $\displaystyle = 0.5 B + 0.2 A$    
$\displaystyle C_$new $\displaystyle = 0.9 C + 0.2 B$    
$\displaystyle D_$new $\displaystyle = 0.1 D + 0.6 C + 0.2 E$    
$\displaystyle E_$new $\displaystyle = 0.8 E + 0.3 B + 0.1 A\,.$    

Let $ x = (A,B,C,D,E)^{\operatorname t}$, then we obtain:

$\displaystyle x_$new$\displaystyle = \begin{pmatrix} 0.7 & 0 & 0.4 & 0.8 & 0 \\ 0.2 & 0.5 & 0 & 0... ...\\ 0 & 0 & 0.6 & 0.1 & 0.2 \\ 0.1 & 0.3 & 0 & 0 & 0.8 \end{pmatrix} x\,. $

Thus, multiplication by the $ n$-th power of the iteration matrix yields the market shares after $ n$ years.

This iteration converges to eigenvector $ x_$max corresponding to the eigenvalue of maximal modulus $ \lambda_$max. We obtain $ \lambda_$max$ =1.1$ and find the corresponding eigenvector

$\displaystyle x_$max$\displaystyle =(0.75, 0.25, 0.25, 0.25, 0.5)^{\operatorname t}\,. $

In the long run the market will be dominated by enterprise $ A$. The percentage market shares can be found by normalisation $ x_$max$ /\Vert x_$max$ \Vert _1$:

$\displaystyle A:\ 37.5\%,\ B,C,D:\ 12.5\%,\ E:\ 25\% \,. $

(Authors: App/Burkhardt/Höllig)
  automatically generated 4/21/2005