QR Decomposition Calculator

QR decomposition (also called QR factorization) breaks a matrix A into the product of two matrices: Q (an orthogonal matrix whose columns are orthonormal) and R (an upper triangular matrix), so that A = QR. It is widely used in numerical linear algebra for solving linear systems, computing eigenvalues, and performing least-squares fitting.

This calculator uses the classical Gram-Schmidt orthogonalization process to compute the Q and R matrices. Starting from the columns of A, it iteratively orthogonalizes and normalizes each vector to build Q, then recovers R as R = Qᵀ·A. Note that classical Gram-Schmidt can lose numerical precision for nearly dependent columns; modified Gram-Schmidt or Householder reflections are more stable for production use.

The columns of the input matrix must be linearly independent. If two or more columns are linearly dependent (or nearly so), the Gram-Schmidt process will encounter a zero-norm vector, making it impossible to normalize. The calculator will alert you if this condition is detected.

Q is an orthogonal matrix, meaning its columns are mutually orthogonal unit vectors. As a result, Qᵀ·Q = I (the identity matrix), and the inverse of Q equals its transpose. This makes Q very useful in numerical computations because it preserves vector norms.

R is an upper triangular matrix — all entries below the main diagonal are zero. The diagonal entries of R equal the norms of the intermediate (un-normalized) orthogonalized vectors from the Gram-Schmidt process. If A has full column rank, all diagonal entries of R are nonzero.

Multiply Q by R and confirm you recover the original matrix A. The calculator provides a verification norm ||A − QR||, which should be extremely close to zero (within floating-point precision, typically less than 1e-10). The column norms of Q should all equal 1, and Q·Qᵀ should approximate the identity matrix.

QR decomposition is used for solving least-squares problems, computing the eigenvalues of a matrix via the QR algorithm, solving linear systems more stably than LU decomposition in some cases, and performing principal component analysis (PCA). It is a foundational tool in scientific computing, statistics, and machine learning.

LU decomposition factors A into a lower triangular matrix L and upper triangular matrix U, and is primarily used for solving square linear systems. QR decomposition factors A into an orthogonal matrix Q and upper triangular matrix R, which is more numerically stable and applies to rectangular matrices. QR is preferred when solving least-squares problems or when orthogonality properties are needed.