Matrix Factorization via Gram-Schmidt

Linear Algebra (Matrices) · Axiom Academy

Run Gram-Schmidt on the columns of a matrix and the factorization A = QR falls out — the orthonormal vectors become Q , and the projection coefficients become R . 1. The Columns Become Orthonormal Read A column by column : and . Gram-Schmidt peels these apart into unit vectors that are perpendicular to each other — the columns q_1, q_2 of Q . Press play to watch a_1 shrink to the unit vector q_1 , then a_2 lose its q_1 -component so the leftover lifts off perpendicular and normalizes to q_2 . Subtract the q_1 -part, then normalize the remainder Each entry of R is a single dot product: . Press play to fill R one cell at a time — the diagonal entries are the remainder lengths , the above-diagonal entry is the projection , and the cell below the diagonal stays 0 because q_2 was built perpendicular to a_1 . and — the size of each remainder before it was normalized. — how much of a_2 points along q_1 . . Since , the dot product vanishes. q_i depends only on , so whenever i > j — every below-diagonal entry is 0 . Reverse the bookkeeping: each column of A is a combination of the q 's with the r 's as weights. Press play to rebuild a_2 tip-to-tail as — it lands exactly on — and . Column j uses only , which is precisely why R is upper triangular and why QR reproduces A exactly. The orthonormal half: holds the directions, with . The triangular half: holds the lengths and overlaps, with a 0 below the diagonal.

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