QR Decomposition Calculator

QR decomposition factors a matrix A into A = QR, where Q has orthonormal columns (Q^T Q = I, the identity) and R is upper triangular. It is the workhorse of least-squares regression and the engine behind the QR algorithm for finding eigenvalues.

This calculator uses classical Gram-Schmidt orthogonalization on the columns of A:

For each column j: take the original column, subtract its projections onto all previously orthonormalized columns, then normalize. The lengths of the original-minus-projections become the diagonal of R, and the projection coefficients fill in the off-diagonal entries of R.

Once you have Q and R, solving Ax = b becomes Rx = Q^T b — a triangular system, fast to back-substitute. For least-squares (overdetermined Ax ≈ b), the same idea: R x = Q^T b gives the least-squares solution directly, no normal equations needed. Numerical analysts strongly prefer QR over solving A^T A x = A^T b because the normal equations square the condition number and can blow up for nearly singular A.

Modified Gram-Schmidt and Householder reflections give better numerical stability than classical Gram-Schmidt, but the classical version is easiest to implement and shows the algorithm clearly. For the small (up to 4x4) matrices supported here, classical Gram-Schmidt is fine.

Worked example: A = [[12,-51,4],[6,167,-68],[-4,24,-41]]. The first column has length sqrt(196) = 14. After orthonormalizing all three columns and forming R from the projections, you get an orthogonal Q and an upper-triangular R whose product equals A.

R inherits a useful property: |det(A)| = |det(R)| = product of |diagonal entries of R|. Since R is upper triangular, this is just the product of its diagonal. Q has determinant ±1 (orthogonal matrices preserve volume).

If A is singular or has linearly dependent columns, classical Gram-Schmidt fails when one of the column lengths after subtraction comes out to zero. The calculator returns an error in that case.