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The $QR$ factorization provides us with a way to write every real matrix $A$ in the form of $QR$, with $Q$ being an orthogonal matrix and $R$ being an upper triangular matrix. I believe that there should also exist a complex version of the $QR$ factorization.

On the other hand, the Schur decomposition provides us a way to write every complex matrix $A$ in the form of $QUQ^{-1}$, where $Q$ is an unitary matrix and $U$ is an upper triangular matrix. Note that the notion of unitary matrices is just a complex generalization of orthogonal matrices.

I feel like the $QR$ factorization is better than the Schur decomposition, because in the first decomposing way we only have two factors which make the expression neater. But I think now that the Schur decomposition exists, there must be some meanings of its, which I can not see.

Can anyone reveal any possible connection between them to me? Thanks for help.

Rodrigo de Azevedo
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Sam Wong
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    I came up with an advantage of the Schur decomposition. If we want to calculate the powers of a complex matrix, then the Schur decomposition makes the calculation simpler while the QR factorization doesn't. Any other idea or comment is still welcome. – Sam Wong May 02 '21 at 12:54
  • The Schur decomposition provides a unitary similarity to a triangular matrix, whereas the QR does not. Similarities are important, because they can review basis invariant properties about the underlying operator a matrix represents. As a consequence, many important results in matrix analysis can be derived from Schur's theorem, so it's a theoretically significant result (see chapters 1-3 from Horn's Matrix Analysis). – V.S.e.H. Nov 12 '22 at 23:42

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