Find the QR decomposition of a machine-precision matrix:

Format the result:

QR decomposition for a complex matrix:

Verify the decomposition using ConjugateTranspose:

Use QRDecomposition for an exact matrix:

QR decomposition for an arbitrary-precision matrix:

Use QRDecomposition with a symbolic matrix:

The QR decomposition for a large numerical matrix is computed efficiently:

A rank deficient, square matrix:

QRDecomposition[m] gives a rectangular matrix:

QRDecomposition[m,Full] adds additional rows, q square and r the same shape as m:

Both pairs are valid decompositions:

The leading rows of the two decompositions are the same:

The r matrix is always padded with zero rows:

The full decomposition always adds rows when the input is tall:

This is because a tall matrix can never have full row rank:

For wide matrix m, the full and thin decompositions coincide:

This is because it has full row rank:

For the wide matrix d, the two decompositions are not the same:

This is because d does not have full row rank:

Find the QR decomposition for a sparse matrix:

QR decompositions of structured matrices:

Use with a QuantityArray structured matrix that has consistent units:

The matrix is dimensionless; the matrix gets the units:

QRDecomposition[m] when m is an identity matrix consists of two identity matrices:

QR decomposition of HilbertMatrix:

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:

With default settings, QRDecomposition returns a list of two dense matrices for real input:

With TargetStructure->"Structured", the result is a pair of OrthogonalMatrix and UpperTriangularMatrix objects:

With the default setting of TargetStructure, the pivot matrix is returned as a SparseArray object:

Use TargetStructure"Dense" to get all three matrices as ordinary lists of lists:

With the setting TargetStructure->"Structured", the result for real input is a triple of OrthogonalMatrix, UpperTriangularMatrix and PermutationMatrix objects:

For complex input and TargetStructure->"Structured", QRDecomposition returns a pair of UnitaryMatrix and UpperTriangularMatrix objects:

Get the result as sparse matrices instead:

Find an orthonormal basis for the column space of the following matrix , and then use that basis to find a QR factorization of :

Compute the dimensions of :

Define as the column of and as the element of the corresponding Gram–Schmidt basis:

Let be the matrix whose rows are the :

Let be the matrix whose elements are the components of along the basis vector:

Confirm that :

This is the same result as given by QRDecomposition:

Compare QR decompositions found using Orthogonalize and QRDecomposition for the following matrix :

Let be the result of applying Orthogonalize to the columns of :

Let equal :

Confirm that :

This is the same result as given by QRDecomposition:

Compare QR decompositions found using Orthogonalize and QRDecomposition for the following matrix :

Let be the result of applying Orthogonalize to the complex-conjugated columns of :

Let equal :

Confirm that :

Up to sign, this is the same result as given by QRDecomposition:

For some applications, it use useful to compute a so-called full QR decomposition, in which the is square (and thus unitary) and has the same dimensions as the input matrix. Compute the full QR decomposition for the following matrix :

There are only two linearly independent columns, so and each have only two rows:

Use NullSpace to find vectors outside the span of the rows of , then orthogonalize the complete set:

This matrix is unitary:

Simply pad the matrix with zeros to make it the same shape as :

Verify that this is also a valid QR decomposition:

Use the QR decomposition to find the that minimizes for the following matrix and vector :

Compute decomposition of :

Since , , and the normal equations can be recast as :

As is invertible (because the columns of are linearly independent), the solution is :

Confirm the result using LeastSquares:

Use the QR decomposition to solve for the following matrix and vector :

Compute the QR decomposition of , which gives an invertible , as has linearly independent rows:

Let as if solving the least-squares problem:

As the columns of span , must be a solution of the equation:

QRDecomposition can be used to find a best-fit curve to data. Consider the following data:

Extract the and coordinates from the data:

Let have the columns and , so that minimizing will be fitting to a line :

As the columns of are linearly independent, the coefficients for a linear least‐squares fit are :

Verify the coefficients using Fit:

Plot the best-fit curve along with the data:

Find the best-fit parabola to the following data:

Extract the and coordinates from the data:

Let have the columns , and , so that minimizing will be fitting to :

As the columns of are linearly independent, the coefficients for a least‐squares fit are :

Verify the coefficients using Fit:

Plot the best-fit curve along with the data: