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i am interested in finding eigenvalues and eigenvectors for any real matrix - large and not symmetric. i discovered that QR-algoritm is the best one. I have questions:

1) if A-matrix (real and not symmetric) is given, then $Q$-matrix can be found for example using Gramm-Schmidt orthogonalization procedure - if $A$ is real then in any case $Q$ is real too ... and $R$-matrix is real either (do I understand correctly ?)

2) if $A$ is real and not symmetric then the eigenvalues can be complex numbers. then, my question is - how to use $QR$-algorithm in order to find complex eigenvalues too ? (if the sequence $A_{(k+1)}=R_kQ_K$ will produce real matrices $A_{(k+1)}$ ?)

Mini_me
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R.Goldov
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  • Good question. As an example, what happens if $A$ is already orthogonal? In this case one $QR$ factorization has $Q=A$ and $R=I$, so the QR algorithm doesn't seem to do anything at all. Have I made a mistake? – felipeh Aug 28 '17 at 04:18
  • If the QR-algorithm is applied correctly, the matrix $A_n$ converges towards a matrix with $2\times 2$ diagonal blocks, from which the complex eigenvalues can easily be derived. – Reinhard Meier Aug 28 '17 at 08:10
  • @felipeh You need shifts to make it work. – Algebraic Pavel Aug 30 '17 at 22:33

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