Can we make an algorithm for Schur Decomposition?

Asked Jan 21 '20 at 14:58

Active Jan 22 '20 at 21:51

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Schur's Theorem If $A$ is any $n \times n$ complex matrix, there exist a unitary matrix $U$ such that $$U^*AU=T$$ is upper triangular. Moreover, the entries on the main diagonal of $T$ are the eigenvalues of $A$.

I've read the proof in W. Keith Nicholson's book Linear Algebra with Applications and I don't know how to make an algorithm from it since the proof is by induction. So, is there any way to find the unitary matrix $U$ satisfying the Schur's Theorem?

edited Jan 22 '20 at 21:51

Rodrigo de Azevedo

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asked Jan 21 '20 at 14:58

RANGGAJAYA CIPTAWAN

See e.g. Wikipedia – Robert Israel Jan 21 '20 at 15:11
I still can't find the use of the QR decomposition. Can you explain it briefly? – RANGGAJAYA CIPTAWAN Jan 21 '20 at 15:33
It's the QR algorithm, not the QR decomposition. Start with $A_0 = A$; given $A_k$, write the QR decomposition $A_k = Q_k R_k$ and take $A_{k+1} = R_k Q_k$. The sequence $A_k$ converges to $T$ (under appropriate conditions). Note that $A_k = Q_k A_{k+1} Q_k^*$, so $U = \lim_{k \to \infty} Q_0 Q_1 \ldots Q_k$. – Robert Israel Jan 21 '20 at 22:19
Hmm..so it's not that simple. But thank you for the explanation. – RANGGAJAYA CIPTAWAN Jan 22 '20 at 00:34
In Wikipedia the QR algorithm is stated only for real matrices. Where can I find a programmable algorithm for Schur decomposition for the general case? – Triceratops May 20 '20 at 12:20

Can we make an algorithm for Schur Decomposition?

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