QR Decomposition Calculator

About this calculator

QR decomposition factors a matrix A into the product of an orthogonal matrix Q (whose columns are unit vectors, mutually perpendicular) and an upper-triangular matrix R. For a 2×2 matrix, this calculator applies the Gram-Schmidt process to the column vectors: the first column is normalized to form Q's first column, then the second column has its projection onto the first removed and is normalized to form the second — the projection coefficients and remaining lengths become the entries of R.

QR decomposition is a workhorse of numerical linear algebra, underpinning the least-squares method used to fit regression lines and curves to data, and forming the basis of the QR algorithm used to compute eigenvalues of a matrix. It's also more numerically stable than solving normal equations directly, which is why statisticians, engineers, and scientific-computing libraries such as LAPACK use it for solving linear systems and fitting models in practice.