QR Factorization

Compute the QR Factorization¹ of a matrix \(A\) using Householder reflections².
Useful for solving least squares problems and computing eigenvalues.

For an \(m \times n\) matrix \(A\), the QR factorization is:

\[ A = QR \]

Components:

\(Q\) is an \(m \times m\) orthogonal matrix (or \(m \times n\) for economy QR)
\(R\) is an \(m \times n\) upper triangular matrix (or \(n \times n\) for economy QR)
\(Q^TQ = I\) (columns of \(Q\) are orthonormal)

Applications:

Solving least squares problems³: \(\min \|Ax - b\|\)
Computing eigenvalues (QR algorithm)⁴
Orthonormalizing a set of vectors

This calculator uses Householder reflections to compute the factorization, with options for full or economy QR and decimal or surd output formats.

Rows \(m\): Columns \(n\):

Enter dimensions and click Generate Matrix to create the input grid.

QR Type: Full Economy

Output Form: Decimal Surd

Set Up

Loads a pre-filled 3×2 matrix — ready to compute.

Compute

Decomposes the matrix into an orthogonal Q and upper-triangular R via Gram–Schmidt.

Resets all dimensions, matrix entries, and results.

Footnotes

The QR factorization decomposes a matrix \(A\) into a product \(A = QR\), where \(Q\) is orthogonal and \(R\) is upper triangular. It is one of the most important matrix decompositions in numerical linear algebra. ↑
Householder reflections are orthogonal transformations of the form \(H = I - 2vv^T\) that zero out entries below the diagonal. They are numerically more stable than Gram-Schmidt orthogonalization. ↑
The least squares problem \(\min \|Ax - b\|_2\) can be solved efficiently via QR: since \(Q\) is orthogonal, \(\|Ax - b\| = \|Rx - Q^Tb\|\), reducing the problem to back-substitution with \(R\). ↑
The QR algorithm iteratively computes \(A_k = Q_kR_k\), then forms \(A_{k+1} = R_kQ_k\). Under mild conditions the sequence converges to an upper triangular matrix whose diagonal entries are the eigenvalues of \(A\). ↑