Computing QR Factorization

Linear Algebra (Matrices) · Axiom Academy

EXAMPLE Computing a QR Factorization Run Gram–Schmidt on the columns of a matrix to build Q and R with A = QR . Factor the matrix A as A = QR , where Q has orthonormal columns and R is upper triangular. We will apply the Gram–Schmidt process to the columns of A . Nice work — you factored A into QR by running Gram–Schmidt on its columns. Gram–Schmidt: normalize , then strip each later column of its component along the orthonormal vectors already built, and normalize what is left. The matrix Q : its columns are orthonormal, so Q^ T Q = I . The matrix R : upper triangular, with diagonal entries the norms ( , ) and the off-diagonal entry the projection . QR factorization is a workhorse of numerical linear algebra — it underlies least-squares solving and eigenvalue algorithms.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.