Introduction to Linear Algebra, Fifth Edition, Section 4.4 #28 The power method

Question

Introduction to Linear Algebra, Fifth Edition, Section 4.4 #28

The power method is a numerical method used to estimate the dominant eigenvalue of a matrix A. (By the dominant eigenvalue, we mean the one that is largest in absolute value. ) the algorithm proceeds as follows: Choose any starting vector Let x k + i = Ax k, k = 0, 1, 2, . . . . Let Under suitable conditions, it can be shown that { beta K} rightarrow lambda 1 where Xi is the dominant eigenvalue of A. Use the power method to estimate the domi-nant eigenvalue of the matrix in Exercise 9. Use the starting vector and calculate beta 0, beta 1, beta 2, beta 3, and beta 4

Explanation / Answer

You calculate this by, as the example says, calculating x1=Ax0, x2=Ax1,...x5=Ax4, along with the estimate for the largest eigenvalue, which is xkTxk+1/xkTxk The following table shows the results If you carry this out to additional steps, you will see that B converges to 2. Step 0 1 2 3 4 5 x 1 1 9 17 41 81 y 1 -7 -7 -23 -39 -87 z 1 1 17 33 81 161 xk+1Txk -5 75 875 4267 19755 xkTxk 3 51 419 1907 9763 Bstep-1 -1.6667 1.4706 2.0883 2.2375 2.023455905 The bottom row gives B0, B1, B2, B3, and B4 You will also see that x, y, z converge to the ratio 1, -1, 2 (this is approximately the ratio of 81, -87, 161) If you plug in 1, -1, 2, you will see that this is indeed an eigenvector and it has eigenvalue 2.

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