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The Power Method for Eigenvectors

Power Method

We now describe the power method for computing the dominant eigenpair.  Its extension to the inverse power method is practical for finding any eigenvalue provided that a good initial approximation is known.  Some schemes for finding eigenvalues use other methods that converge fast, but have limited precision.  The inverse power method is then invoked to refine the numerical values and gain full precision.  To discuss the situation, we will need the following definitions.

Definition  If is an eigenvalue of A  that is larger in absolute value than any  other eigenvalue, it is called the dominant eigenvalue. An eigenvector   corresponding to is called a dominant eigenvector.

Definition  An eigenvector  V  is said to be normalized if the coordinate of largest magnitude is equal to unity (i.e., the largest coordinate in the vector  V  is the number 1).

Remark.  It is easy to normalize an eigenvector     by forming a new vector    where    and  .

Theorem (Power Method)  Assume that the n×n matrix  A  has  n  distinct eigenvalues    and that they are ordered in decreasing magnitude; that is,  .  If    is chosen appropriately, then the sequences   and    generated recursively by

and

where
and  ,  will converge to the dominant eigenvector    and eigenvalue  ,  respectively. That is,

and  .

Remark.  If   is an eigenvector and , then some other starting vector must be chosen.

Speed of Convergence

In the iteration in the theorem uses the equation

,

and the coefficient of
that is used to form     goes to zero in proportion to .  Hence, the speed of convergence of   to is governed by the terms  .  Consequently, the rate of convergence is linear.  Similarly, the convergence of the sequence of constants to is linear.  The Aitken method can be used for any linearly convergent sequence to form a new sequence,

,

that converges faster. The  Aitken
can be adapted to speed up the convergence of the power method.

Shifted-Inverse Power Method

We will now discuss the shifted inverse power method.  It requires a good starting approximation for an eigenvalue, and then iteration is used to obtain a precise solution.  Other procedures such as the
QM and Givens’ method are used first to obtain the starting approximations.  Cases involving complex eigenvalues, multiple eigenvalues, or the presence of two eigenvalues with the same magnitude or approximately the same
magnitude will cause computational difficulties and require more advanced methods.  Our illustrations will focus on the case where the eigenvalues are distinct.  The shifted inverse power method is based on the following three results (the proofs are left as exercises).

Theorem (Shifting Eigenvalues)  Suppose that  ,V  is an eigenpair of  A.  If    is any constant, then  ,V  is an eigenpair of the matrix  .

Theorem (Inverse Eigenvalues)   Suppose that  ,V  is an eigenpair of  A.   If  ,  then  ,V  is an eigenpair of the matrix  .

Theorem (Shifted-Inverse Eigenvalues)  Suppose that  ,V  is an eigenpair of  A.   If  ,   then  ,V  is an eigenpair of the matrix  .

Theorem (Shifted-Inverse Power Method)  Assume that the n×n matrix  A  has distinct eigenvalues    and consider the eigenvalue . Then a constant    can be chosen so that    is the dominant eigenvalue of  .  Furthermore, if     is chosen appropriately, then the  sequences   and    generated recursively by

and

where
and  ,  will converge to the dominant eigenpair  ,  of the matrix  .   Finally, the corresponding eigenvalue for the matrix  A  is given by the calculation

Remark.  For practical implementations of this Theorem, a linear system solver is used to compute   in each step by solving the linear system .

Computer Programs  Power Method  Power Method

Mathematica Subroutine (Power Method).  To compute the dominant value    and its associated eigenvector    for the n×n matrix  A.  It is assumed that the n eigenvalues have the dominance property   .

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Example 1.  Use the power method to find the dominant eigenvalue and eigenvector for the matrix  .
Solution 1.

Example 2.  Use the power method to find the dominant eigenvalue and eigenvector for the matrix  .
Solution 2.

Shifted Inverse Power Method

If a good approximation to an eigenvalue is known, then the shifted inverse power method can be used and convergence is faster.  Other methods such as the QM method or Givens method are used to obtain approximate starting values.

Program (Shifted Inverse Power Method).  To compute the dominant eigenvalue    and its associated eigenvector    for the  n by n  matrix  A.  It is assumed that the n eigenvalues are    and  α  is a real number such that    for each  .

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Example 3.  Find the dominant eigenvalue and eigenvector for the matrix  .
Use the shift   in the shifted inverse power method.
Solution 3.

Application to Markov Chains

In the study of Markov chains the elements of the transition matrix are the probabilities of moving from any state to any other state.  A Markov process can be described by a square matrix whose entries are all positive and the column sums are all equal to 1.  For example, a 3×3 transition matrix looks like

where   ,    and  .  The initial state vector is  .

The computation    shows how the   is redistributed in the next state.   Similarly we see that

shows how the   is redistributed in the next state.
and
shows how the   is redistributed in the next state.

Therefore, the distribution for the next state is

A recursive sequence is generated using the general rule

for  k = 0, 1, 2, ... .

We desire to know the limiting distribution  .  Since we will also have    we obtain the relation

From which it follows that

Therefore the limiting distribution  P  is the eigenvector corresponding to the dominant eigenvalue  .  The following subroutine reminds us of the iteration used in the power method.

Mathematica Subroutine (Markov Process).

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Example 4.  Let    record the number of people in a certain city who use brands X, Y, and Z, respectively.
Each month people decide to keep using the same brand or switch brands.
The probability that a user of brand
X will switch to brand Y or Z is 0.3 and 0.3, respectively.
The probability that a user of brand
Y will switch to brand X or Z is 0.3 and 0.2, respectively.
The probability that a user of brand
Z will switch to brand X or Y is 0.1 and 0.3, respectively.
The transition matrix for this process is
or

Assume that the initial distribution .
4 (a).  Find the first few terms in the sequence .
4 (b).  Verify that   is the dominant eigenvector of  A.
4 (c).  Verify that a corresponding eigenvector is  .
4 (d).  Conclude that the limiting distribution is  .
Solution 4.

Example 5.  Let    record the number of people in a certain city who use brands X, Y, and Z, respectively.
Each month people decide to keep using the same brand or switch brands.
The probability that a user of brand
X will switch to brand Y or Z is 0.4 and 0.2, respectively.
The probability that a user of brand
Y will switch to brand X or Z is 0.3 and 0.2, respectively.
The probability that a user of brand
Z will switch to brand X or Y is 0.1 and 0.3, respectively.
The transition matrix for this process is
or

Assume that the initial distribution .
5 (a).  Find the first few terms in the sequence .
5 (b).  Verify that   is the dominant eigenvector of  A.
5 (c).  Verify that a corresponding eigenvector is  .
5 (d).  Conclude that the limiting distribution is  .
Solution 5.

Power Method  Power Method  Internet hyperlinks to web sites and a bibliography of articles.

(c) John H. Mathews 2004