qR Decomposition Preview

Math Linear Algebra • Matrices and Systems of Equations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Factor \(A\) into \(A=QR\) using modified Gram–Schmidt, where \(Q\) has orthonormal columns and \(R\) is upper triangular. Includes checks for orthonormality (\(Q^TQ\approx I\)) and reconstruction (\(QR\approx A\)). Optional solve: \(Ax=b\) (least-squares if \(m>n\)).

Rows (m)

Cols (n)

Display decimals

Tolerance

Steps

Solve \(Ax=b\) (least squares if \(m>n\))

Play animates Gram–Schmidt snapshots once and stops automatically. Drag to pan • wheel to zoom • double-click to reset.

Matrix \(A\)

2×2

Vector \(\mathbf{b}\)

2×1

Ready

Matrix \(Q\)

Orthonormal columns (m×n)

Matrix \(R\)

Upper triangular (n×n)

Check \(Q^TQ\)

Should be close to \(I_n\)

Modified Gram–Schmidt animation

Shows how \(Q\) and \(R\) are built column-by-column. Pivot cell indicates the current \(r_{jk}\) being written.

Q R current entry current column

Enter \(A\) (and optionally \(\mathbf{b}\)) and click “Calculate”.

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Theory

What QR decomposition is

The QR decomposition factors a matrix \(A\in\mathbb{R}^{m\times n}\) (typically with \(m\ge n\)) into \[ A = QR, \] where \(Q\in\mathbb{R}^{m\times n}\) has orthonormal columns and \(R\in\mathbb{R}^{n\times n}\) is upper triangular. Orthonormal columns means \(Q^TQ = I_n\). This makes QR extremely useful in numerical linear algebra because orthonormal transformations preserve lengths and angles, which reduces rounding-error amplification.

Modified Gram–Schmidt construction

One way to compute QR is via (modified) Gram–Schmidt. View \(A\) as a list of column vectors: \(A=[a_1\ a_2\ \dots\ a_n]\). The idea is to orthogonalize and then normalize these columns.

Start with \(v_1=a_1\). Set \(r_{11}=\|v_1\|\) and \(q_1 = v_1/r_{11}\). For the next column, subtract the projection onto \(q_1\): \[ r_{12}=q_1^T a_2,\qquad v_2=a_2-r_{12}q_1, \] then normalize \(v_2\) to get \(q_2\). In general, the modified Gram–Schmidt updates each remaining column after computing \(q_k\): \[ r_{k j} = q_k^T v_j,\qquad v_j \leftarrow v_j - r_{k j} q_k\quad (j>k), \] and finally \(r_{kk}=\|v_k\|\), \(q_k=v_k/r_{kk}\).

If a column becomes nearly zero after subtracting projections, then the original columns were linearly dependent (or nearly so), and \(r_{kk}\) will be close to zero. In that case, QR still provides insight, but the factorization is not unique and solving with \(R\) can become unstable.

Why QR is used for least squares

When \(m>n\), the system \(Ax=b\) is overdetermined. Usually there is no exact solution, so we seek the least-squares solution that minimizes \(\|Ax-b\|\). Using QR: \[ Ax \approx b \quad\Rightarrow\quad QRx \approx b. \] Multiply by \(Q^T\) and use \(Q^TQ=I\): \[ Rx \approx Q^T b. \] This reduces least squares to solving the triangular system \(Rx=y\) where \(y=Q^Tb\). This is generally more stable than forming the normal equations \(A^TAx=A^Tb\), which can square the condition number.

Checks shown by the calculator

The tool verifies two standard properties:

Orthonormality: \(Q^TQ\) should be close to \(I_n\). The calculator reports \(\max |Q^TQ-I|\).
Reconstruction: \(QR\) should be close to \(A\). The calculator reports \(\max |A-QR|\).

Small values indicate that the computed QR is consistent at the chosen tolerance and rounding level.

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