QR decomposition

In linear algebra, a QR decomposition (also called a QR factorization) of a matrix is a decomposition of a matrix A into a product A = QR of an orthogonal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares problem, and is the basis for a particular eigenvalue algorithm, the QR algorithm.

Any real square matrix A may be decomposed as

where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning Q^TQ = I) and R is an upper triangular matrix (also called right triangular matrix). If A is invertible, then the factorization is unique if we require that the diagonal elements of R be positive.

If instead A is a complex square matrix, then there is a decomposition A = QR where Q is a unitary matrix (so Q^*Q = I).

If A has n linearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n. The fact that any column k of A only depends on the first k columns of Q is responsible for the triangular form of R.

...
Wikipedia