# Can QR decomposition be used for matrix inversion?

1. Is there any simple algorithm for matrix inversion (that can be implemented using C/C++)?

2. Can QR decomposition be used for matrix inversion? How?

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Please clarify what is B"H in the beginning of the question? – user21436 Feb 14 '12 at 17:39
It means with g-d's help (without it there shall be no benefit to this question, nor to any of the answers to it). – Yan Raf Feb 14 '12 at 18:14
The first question might be more suitable for scicomp.SE. – user2468 Feb 20 '12 at 21:09

A QR-decomposition can certainly be used for matrix inversion because if $A=QR$ then $A^{-1} = R^{-1} Q^{-1} = R^{-1} Q^{T}$ and $R^{-1}$ is easy to compute because $R$ is triangular.
But consider why you need to invert a matrix. In most cases, you don't: you just need to solve a linear system $Ax=b$. If $A=QR$ then this system is equivalent to $Rx = Q^T b$, which is easy to solve because $R$ is triangular.
Thanks much for this useful answer. Could you elaborate on "$R^{-1}$ is easy to compute because $R$ is triangular"? How is a triangular matrix easily inverted? – Yan Raf Feb 14 '12 at 18:18
@YanRaf, $R^{-1}$ is easy to invert, since the diagonal contains the eigenvalues from which you get the eigendecomposition. However, maybe lhf was refering to solving $Rx=y$, which is solvable by back-/foward-substitution. – Peter Sheldrick Feb 14 '12 at 18:32