http://en.wikipedia.org/wiki/QR_algorithm
is it possible to write it in one equation
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2You mean you don't want to write $A = QR$, but want a shorthand for it? – Amzoti Feb 01 '13 at 02:27
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i mean Q = something, R = something, rather than an algorithm – Machine Gun Feb 01 '13 at 02:44
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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 01 '13 at 02:53
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after read your comment, it sounds right. – Machine Gun Feb 01 '13 at 03:15
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still hope another way to do this algorithm – Machine Gun Feb 01 '13 at 03:24
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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.
user7530
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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 01 '13 at 04:20
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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 01 '13 at 04:24