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http://en.wikipedia.org/wiki/QR_algorithm

is it possible to write it in one equation

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You mean you don't want to write $A = QR$, but want a shorthand for it? –  Amzoti Feb 1 '13 at 2:27
    
i mean Q = something, R = something, rather than an algorithm –  Machine Gun Feb 1 '13 at 2:44
    
Interesting question, though any answer would be far more conceptual than anything. $A=QR$ gives $Q=AR^{-1}$ and $R=Q^{-1}A$, but that is about it. To find orthogonal $Q$ and triangular $R$ for $A=QR$ would require an algorithm, not a simple formula/equation. So I guess the answer is no. –  adam W Feb 1 '13 at 2:53
    
after read your comment, it sounds right. –  Machine Gun Feb 1 '13 at 3:15
    
still hope another way to do this algorithm –  Machine Gun Feb 1 '13 at 3:24
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1 Answer

up vote 1 down vote accepted

Are you asking about the QR algorithm or the QR decomposition?

The factors $Q$ and $R$ can indeed be written using explicit formulas, since the columns of $Q$ are given by the Gram-Schmidt process, and can thus be expressed by (increasingly complicated) functions of the entries of $A$ that are continuous away from $\det A =0$. As adam points out, once you have $Q$, you also have $R$.

The output of the $QR$ algorithm is the spectrum of $A$. It does have a formula, in a sense, given by the characteristic polynomial; an explicit formula for the eigenvalues of $A$ would also be an explicit formula for the roots of all monic polynomial. To my knowledge no such formula exists, and by the Abel-Ruffini theorem it cannot be algebraic.

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does it mean all decomposition algorithms in matrix also do not have explicit formula? if have explicit formulas, which of them? –  Machine Gun Feb 1 '13 at 4:20
    
That depends on the decomposition. Like I say in the first part of my answer, the $QR$ decomposition has a formula. So do the $LU$ and Cholesky decompositions. SVD on the other hand does not. –  user7530 Feb 1 '13 at 4:24
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