Question

The power method for finding eigenvalues. Using technology, generate a random $\mathbf{5}\mathbf{\times }\mathbf{5}$matrix $\mathit{A}$with nonnegative entries. (Depending on the technology you are using, the entries could be integers between zero andnine, or numbers between zero and one.) Using technology, compute$\mathit{B}\mathbf{=}{\mathit{A}}^{\mathbf{20}}$(or another high power of ). We wish to compare the columns of . This is hard to do by inspection, particularly because the entries of$\mathit{B}$may get rather large.

To get a better hold on, form the diagonal$\mathbf{5}\mathbf{\times }\mathbf{5}$matrix $\mathit{D}$whose$\mathit{i}\mathbf{}\mathit{t}\mathit{h}$diagonal element is ${\mathit{b}}_{\mathbf{1}\mathbf{i}}$, the $\mathit{i}\mathbf{}\mathit{t}\mathit{h}$entry of the first row of .$\mathit{B}$ Compute$\mathit{C}\mathbf{=}\mathit{B}{\mathit{D}}^{\mathbf{-}\mathbf{1}}$

a. How is$\mathit{C}$ obtained from? Give your answer in terms of elementary row or column operations.

b. Take a look at the columns of the matrix $\mathit{C}$you get. What do you observe? What does your answer tell you about the columnsof?

c. Explain the observations you made in part b. You may assume that A has five distinct (complex) eigenvalues and that the eigenvalue with maximal modulus is real and positive. (We cannot explain here why this will usually be the case.)

d. Compute. What is the significance of the entries in the top row of this matrix in terms of the eigenvalues of? What is the significance of the columns of in terms of the eigenvectors of ?

Short answer

(a) The matrix for$C=B{D}^{-1}$, it must use$c_{i,j}=\frac{b_{i,j}}{b_{1,j}}$to obtain from$B$.

(b) At the columns of the matrix,$C$it is observed that the$j-th$

1. The columns ofare the eigenvectors ofwith the corresponding eigenvalues being their entries in first row.

Step-by-step solution

Define matrix:

A table of numbers and symbols arranged in the form of rows and columns is known as matrix.

(a) Obtain Cfrom Bby finding the diagonal entry:

Since,$D$is the diagonal matrix whose $j-th$diagonal entry is, thenmust also be a diagonal matrix and its diagonal entry will be.

Hence, to obtain,, it must apply,

Therefore,must be applied to obtainfrom

(b) Find the observation at the columns of thematrix :

Consider the equation obtained in part (a),

$c_{i,j}=\frac{b_{i,j}}{b_{1,j}}$

Hence, it is concluded that the $j-th$column of $C$is$\frac{1}{b_{1,j}}$ times the $j-th$column of$B$

(c) Assume that matrix Ahas five distinct eigenvalues and explain your observation:

Assume${\lambda }_i$,is an eigenvalue of ,$A$ with the corresponding eigenvector ,$x_i$ then ${\lambda }_i^{20}$ an eigenvalue of $B=A^{20}$with the eigenvector.$x_i$

Thus, the eigenvalues of$D^{-1}$ aredata-custom-editor="chemistry" $\frac{1}{b_{1,j}},$ with the corresponding eigenvectors$e_j,1,2,3,4,5$, .

Hence, from part (b) we can conclude,

$\det C=\left(\prod _{j=1}^5\frac{1}{b_{1,j}}\right)\det B$

Express the equation as,

$x_i=\sum _{j=1}^5{\alpha }_{i,j}{e}_j,i=1,2,3,4,5$

Therefore, we have

${\lambda }_i^{20}{x}_i=B{x}_i=B\left(\sum _{j=1}^5{\alpha }_{i,j}{e}_j\right)=B\left(\sum _{j=1}^5\frac{{\alpha }_{i,j}}{b_{1,j}}{D}^{-1}{e}_j\right)$

Thus, from all the given propertiesof matrices, it is concluded that, the eigenvectors of $C$must be same as the eigenvectors of.$B$

(d) Find the significance of the entries in the top row of this matrix by computing the value of:AC

Hence, the $i-th$of$AC$ is,

$\sum _{k=1}^5{a}_{i,k}{c}_{k,j}=\sum _{k=1}^5\frac{a_{i,k}{b}_{k,j}}{b_{1,j}}=\frac{1}{b_{1,j}}\sum _{k=1}^5{a}_{i,k}{b}_{k,j}$

Thus, the $j-th$column of $AC=A^{21}{D}^{-1}$is$\frac{1}{b_{1,j}}$ times the $j-th$column of.$AB=A^{21}$

Since, it is observed that columns of $AC$are eigenvectors of $A$with the corresponding eigenvalues being their corresponding entries in the top row.

Therefore, the columns of$AC$are the eigenvectors of$A$with the corresponding eigenvalues being their entries in first row.