Let α = λ/(v'v), with λ being A's largest eigenvalue and v its corresponding eigenvector, how do I prove that the largest absolute eigenvalue of A − αvv' equals the second largest eigenvalue of A by use of the power method? I think that αvv' is an ortogonal projection matrix but what is the use of multiplying it by λ and subtracting from A?

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    $\begingroup$ Try looking at what this does in the case that $A$ is diagonal, and I think it should then be clear why this works in general $\endgroup$ – Ben Grossmann Oct 31 '18 at 18:03
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    $\begingroup$ Do you assume $A$ is square and diagonalizable? $\endgroup$ – abolfazl Oct 31 '18 at 19:07

I assume $A$ is an $n\times n$ square matrix having $n$ eigenvectors. Therefore, $A = VDV^{-1}$ where I assume columns of $V$ are orthogonal to each other but, they are not normalized. We can write $A$ equivalently as $A = \lambda_1v_1v_1^\top+\sum_{i=2}^n\lambda_iv_iv_i^\top$. Let $B = A-\alpha v_1v_1^\top$ where $\alpha = \frac{\lambda_1}{\|v_1\|_2^2}$. Then, $B = \lambda_1(1-\frac{1}{\|v_1\|_2^2})v_1v_1^\top+\sum_{i=2}^n\lambda_iv_iv_i^\top$. So, if $$\lambda_1(1-\frac{1}{\|v_1\|_2^2}) > \lambda_2,$$ the largest absolute eigenvalue of $B = A-\alpha v_1v_1^\top$ DOES NOT equal the second largest eigenvalue of $A$. You can easily construct many counterexamples. If you have MATLAB, the following code generates a random counterexample:

clear; clc;
Q = randn(3); Q = orth(Q)*diag([3,2,1])
D = diag([200,1.5,1])
A = Q*D*Q^(-1)
a = 2/norm(Q(:,1))^2
B = A-a*Q(:,1)*Q(:,1)'

If the eigenvectors where normalized, $\lambda_1(1-\frac{1}{\|v_1\|_2^2}) = 0$ and you could easily see $\lambda_2$ is the largest eigenvalue of $B$.

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  • $\begingroup$ That’s great, thank you! How could I, without using matlab, show that the last term is not greater than lambda2, so that the power method would indeed return the second greatest? $\endgroup$ – Walter Nap Oct 31 '18 at 20:05

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