QR Decomposition is a matrix factorization method in linear algebra that orthogonally decomposes any $m \times n$ rectangular real or complex matrix into two constituent matrices: an orthogonal matrix $Q$ and an upper triangular matrix $R$. This fundamental theorem relies on the Gram-Schmidt process or Householder transformations to ensure numerical stability while preserving geometric relationships such as inner products and orthogonality. The factorization provides a unique structural representation applicable across disciplines requiring robust solutions for overdetermined, underdetermined, or full-rank systems without relying on explicit matrix inversion.
Factors a matrix with linearly independent columns into Q with orthonormal columns times upper-triangular R, built from Gram-Schmidt.