QRDecomposition

QRDecomposition[m]

yields the QR decomposition for a numerical matrix m. The result is a list {q,r}, where q is a unitary matrix and r is an uppertriangular matrix.

Details and Options

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:

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)

Pivoting  (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 p 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 :

This finds a vector such that 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 ConjugateTranspose[q].r:

Wolfram Research (1991), QRDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/QRDecomposition.html.

Text

Wolfram Research (1991), QRDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/QRDecomposition.html.

BibTeX

@misc{reference.wolfram_2020_qrdecomposition, author="Wolfram Research", title="{QRDecomposition}", year="1991", howpublished="\url{https://reference.wolfram.com/language/ref/QRDecomposition.html}", note=[Accessed: 18-January-2021 ]}

BibLaTeX

@online{reference.wolfram_2020_qrdecomposition, organization={Wolfram Research}, title={QRDecomposition}, year={1991}, url={https://reference.wolfram.com/language/ref/QRDecomposition.html}, note=[Accessed: 18-January-2021 ]}

CMS

Wolfram Language. 1991. "QRDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/QRDecomposition.html.

APA

Wolfram Language. (1991). QRDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/QRDecomposition.html