Built-in Mathematica Symbol

QRDecomposition

QRDecomposition[m]
yields the QR decomposition for a numerical matrix m. The result is a list {q, r}, where q is an orthogonal matrix and r is an upper-triangular matrix.

MORE INFORMATION

The original matrix m is equal to Conjugate[Transpose[q]].r. »

For non-square matrices, q is row orthonormal. »

The matrix r has zeros for all entries below the leading diagonal. »

QRDecomposition[m, Pivoting->True] yields a list {q, r, p} where p is a permutation matrix such that m.p is equal to Conjugate[Transpose[q]].r. »

EXAMPLES

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Basic Examples (2)

Compute the QR decomposition for a 3×2 matrix with exact values:

Compute the QR decomposition for a 2×3 matrix with approximate numerical values:

Compute the QR decomposition for a 3×2 matrix with exact values:

Compute the QR decomposition for a 2×3 matrix with approximate numerical values:

Scope (2)

m is a 3×4 matrix:

QR decomposition computed with exact arithmetic:

QR decomposition computed with machine arithmetic:

QR decomposition computed with 24-digit arithmetic:

QR decomposition for a 3×3 matrix with random complex entries:

Options (1)

Compute the QR decomposition using machine arithmetic with pivoting:

The elements along the diagonal of r are in order of decreasing magnitude:

The matrix

is a permutation matrix:

QRDecomposition satisfies m.p

ConjugateTranspose[q].r:

Applications (1)

Here is some data:

is a design matrix for fitting with basis functions

,

:

Find the QR decomposition of m:

This finds a vector x such that

m.x - b

₂ is a minimum:

These are the coefficients for the least-squares fit:

Properties & Relations (1)

m is a 3×4 matrix:

Compute the QR decomposition:

The rows of q are orthonormal:

r is upper triangular:

m is equal to Conjugate[Transpose[q]].r: