QR

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matrix_qr

QR decomposition A = Q·R with orthonormal Q and upper-triangular R

Signature

Inputs

aSignal|Arrayrequired— A 2-D matrix (may be rectangular).

Outputs

qMatrix— The orthonormal factor $Q$ ($Q^\top Q = I$).
rMatrix— The upper-triangular factor $R$.

Description

QR decomposes a matrix as $A = Q R$ , where $Q$ has orthonormal columns ( $Q^{⊤} Q = I$ ) and $R$ is upper-triangular, using the MKL LAPACK backend (Householder reflections). It is the numerically stable backbone of least-squares fitting, orthogonalization, and iterative eigen-algorithms.

Both factors are dimensionless. QR is the preferred route for solving over-determined systems when you want an explicit orthonormal basis, and for Gram-Schmidt-style orthogonalization done stably.

Mathematics

A = Q R, Q^{⊤} Q = I, R upper-triangular

Examples

Orthonormal basis for a column space

Feed a tall $m \times n$ matrix of (linearly independent) columns into a. The q output's columns form an orthonormal basis for the column space, and r records the upper-triangular coefficients that reconstruct the original columns as $A = QR$ .

Applications

Stable least-squares solving and Gram-Schmidt orthogonalization.
Building orthonormal bases for subspaces / column spaces.
The core step of QR-iteration eigenvalue algorithms.
Conditioning-friendly alternative to the normal equations.

Neat

Householder QR is backward-stable, so it orthogonalizes columns far more reliably than classical Gram-Schmidt.

Because $Q$ is orthonormal, $R$ carries all the scaling — a compact, upper-triangular summary of the original columns.

Known issues

Sign conventions of $Q$ columns / $R$ diagonal are not unique; a valid decomposition may differ in sign across platforms.

For rank-deficient inputs the trailing $R$ rows become near-zero — QR alone does not rank-reveal without column pivoting.