# QR factorization

**QR factorization** A form of matrix factorization widely used in numerical linear algebra. For *A*, an *m* × *n*, *m* ≥ *n*, real square matrix, the factorization takes the form *A* = *QR*

where *Q* is an *m* × *m* orthogonal matrix and *R* is an *m* × *n* matrix whose first *n* rows form an upper (or right) triangular matrix. An important application is in solving overdetermined linear systems of equations of the form *Ax* = *b*, *m* > *n*; *b* is an *m*-component column vector and *x* is a column vector of *n* unknowns. The *QR* factorization, under appropriate conditions, reduces the problem to solving a simpler square upper triangular system of the form *Rx* = *c*.

For a square matrix, *m* = *n*, a further major application is in computing the eigenvalues and eigenvectors of *A*. Here a sequence of *QR* factorizations are carried out in an iteration scheme that ultimately reduces *A* to a matrix of a particularly simple form whose eigenvalues are the same as those of *A*. The eigenvalues (and if required, eigenvectors) can now be easily computed.

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**QR factorization**