Inverse of matrix with QR method
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As far as I recall, the QR decomposition stage requires $O\left(n^3\right)$ operations.
If I assume that you are not finding the inverse, but solving the linear system $Ax=b$, then once you have found the QR decomposition of $A$, the remaining operations are all $O\left(n^2\right)$.
If you are explicitly finding the inverse, then this will require an additional $O\left(n^3\right)$ operations once you have found the QR decomposition.
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Comments
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Amir over 1 year
What is the complexity of finding the inverse of matrix by QR decomposition? A is a $n×n$ with full rank.
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Admin about 11 yearsWhat is the size of the input matrix? $n\times n$? Is it full rank?
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Amir about 11 yearsYes it is $n×n$ and full rank
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joriki about 11 yearsThe title and body seem to bear no relation to each other. I suspect that in the body "matrix" was supposed to read "the inverse of a matrix"?
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Amir about 11 yearsYes, you are right @joriki. I wrote it fast.
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joriki about 11 yearsNote that I had also provided the correct articles to make the question grammatical.
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Steve about 11 yearsThis. More or less all dense linear algebra is in $O(n^3)$ field operations, unless you decide to use a funkier matrix multiplication exponent, i.e. Coppsmith-Winograd's.