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To find the Eigenvalue as per QR Factorization, the diagonal values of R are the EigenValues while every column in Q is an EigenVector. However, this does not seem to be the case in a 2x2 matrix.

Consider M = \begin{bmatrix}2&1\\1&2\end{bmatrix}

Then the QR decomposition (see http://comnuan.com/cmnn0100e/) yields

Q = \begin{bmatrix}.89&-.44\\-.44&-.89\end{bmatrix} R = \begin{bmatrix}-2.23&-1.78\\0&1.34\end{bmatrix}

Note, QR = M checks out.

Note as per this the EigenValues are -2.23 and 1.34, while if you do the calculation manually it is easy to see that EigenValues for M are 1 and 3.

More importantly, M.C = $\lambda$.C does not come correct using the QR method. In our case. for $\lambda$ = -2.23, C = \begin{bmatrix}.89\\-.44\end{bmatrix}

but this does not satisfy: M.C = $\lambda$.C

Is QR not suitable for 2x2 matrices, or is my understanding wrong ?

cbelwal
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  • Why do you think that the diagonal entries of $R$ are the eigenvalues of $A$? – amd Sep 04 '17 at 17:43
  • As per the QR Factorization algorithm, once R has converged the diagonal entries of R are the EigenValues of A. Or am I mis-understanding something ? – cbelwal Sep 04 '17 at 19:10
  • QR factorization is one step in the inner loop of that algorithm. You’ve only performed (less than) one iteration of it so far. $R$ is far from diagonal at this point. If you keep going, after several iterations you’ll see $R$ start to converge to $\operatorname{diag}(3,1)$. – amd Sep 04 '17 at 19:23
  • Thank you. I modified the EigenValue algorithm and it now converges to real EigenValues. Having some issues with EigenVectors but I think I can figure that out. Thanks much ! – cbelwal Sep 04 '17 at 23:14

1 Answers1

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Based on what you’ve said in the comments to your question, it looks like you might be confusing the QR algorithm for finding eigenvalues of a matrix with the QR decomposition of a matrix.

Computing the QR decomposition of a matrix is a step in the inner loop of the QR algorithm. In the simplest version of this algorithm, after computing the matrices $Q$ and $R$, you then replace your matrix with $RQ$ and loop. If this process converges, then $R$ converges to a diagonal matrix with the eigenvalues of the original matrix along the diagonal.

So, in computing the QR decomposition of $M$, you’ve only performed part of one iteration of this algorithm. The matrix $R$ that resulted from this isn’t even close to being diagonal, so there’s no reason to expect that its main diagonal elements are the eigenvalues of $M$. If you now compute $RQ$, compute the QR decomposition of that, &c., after a half-dozen iterations or so you’ll see $R$ begin to converge to a diagonal matrix with the values that you expect.

amd
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