Math · Linear Algebra

QR Factorization Calculator

Decompose any square matrix into Q·R using the modified Gram-Schmidt process — Q orthogonal, R upper triangular, with full verification.

Matrix size

Method

Matrix A

Q (orthogonal)

R (upper triangular)

Verification — Q·R should equal A

Gram-Schmidt: vₛ = aₛ − Σ_j<k (qₖ·aₛ) qₖ, then qₛ = vₛ / ‖vₛ‖.
Entries of R: r_j,k = qₖ·aₛ (j < k), r_k,k = ‖vₛ‖. Q^TQ = I and A = QR.

Verify Q^TQ = I with rounded residual displayed.

Modified or classical Gram-Schmidt, selectable.

Q·R reconstructed and compared with the input A.

Per-column projection shown as readable steps.

More Linear Algebra

Explore matrix and linear algebra calculators.