# Power Method Algorithm for Finding Dominant Eigen Value and Eigen Vector

In many real world applications of science and engineering, it is required to find numerically the largest or dominant Eigen value and corresponding Eigen vector. There are different methods like Cayley-Hamilton method, Power Method etc. Out of these methods, Power Method follows iterative approach and is quite convenient and well suited for implementing on computer.

In this article we are going to develop an Algorithm for Power Method for computing largest or dominant Eigen value and corresponding Eigen vector.

Let A be the square matrix of order n i.e. An x n. Then Power Method starts with one initial approximation to Eigen vector corresponding to largest Eigen value of size n x 1. Let this initial approximation be Xn x 1.

After initial assumption, we calculate AX i.e. product of matrix A and X. From the product of AX we divide each element by largest element (by magnitude) and express them as λ1X1. Obtained value of λ1 and X1 are next better approximated value of largest Eigen value and corresponding Eigen vector.

Similarly, for the next step, we multiply A by X1. From the product of AX1 we divide each element by largest element (by magnitude) and express them as λ2X2. Obtained value of λ2 and X2 are next better approximated value of largest Eigen value and corresponding Eigen vector.

And then we repeat this process until largest or dominant Eigen value and corresponding Eigen vector are obtained within desired accuracy.

## Algorithm for Power Method

``` 1. Start

2. Read Order of Matrix (n) and Tolerable Error (e)

3. Read Matrix A of Size n x n

4. Read Initial Guess Vector X of Size n x 1

5. Initialize: Lambda_Old = 1

6. Multiply: X_NEW = A * X

7. Replace X by X_NEW

8. Find Largest Element (Lamda_New) by Magnitude from X_NEW

9. Normalize or Divide X by Lamda_New

10. Display Lamda_New and X

11. If |Lambda_Old - Lamda_New| > e then
set Lambda_Old = Lamda_New and goto
step (6) otherwise goto step (12)

12. Stop
```