If A has nlinearly independent columns, then the first n columns of Q form an orthonormal basis for the column space of A. More generally, the first k columns of Q form an orthonormal basis for the span of the first k columns of A for any 1 ≤ k ≤ n. The fact that any column k of A only depends on the first k columns of Q corresponds to the triangular form of R.
More generally, we can factor a complex m×n matrix A, with m ≥ n, as the product of an m×munitary matrixQ and an m×n upper triangular matrix R. As the bottom (m−n) rows of an m×n upper triangular matrix consist entirely of zeroes, it is often useful to partition R, or both R and Q:
where R1 is an n×n upper triangular matrix, 0 is an (m − n)×n zero matrix, Q1 is m×n, Q2 is m×(m − n), and Q1 and Q2 both have orthogonal columns.
Golub & Van Loan (1996, §5.2) call Q1R1 the thin QR factorization of A; Trefethen and Bau call this the reduced QR factorization. If A is of full rankn and we require that the diagonal elements of R1 are positive then R1 and Q1 are unique, but in general Q2 is not. R1 is then equal to the upper triangular factor of the Cholesky decomposition of A*A (= ATA if A is real).
The RQ decomposition transforms a matrix A into the product of an upper triangular matrix R (also known as right-triangular) and an orthogonal matrix Q. The only difference from QR decomposition is the order of these matrices.
QR decomposition is Gram–Schmidt orthogonalization of columns of A, started from the first column.
RQ decomposition is Gram–Schmidt orthogonalization of rows of A, started from the last row.
The Gram-Schmidt process is inherently numerically unstable. While the application of the projections has an appealing geometric analogy to orthogonalization, the orthogonalization itself is prone to numerical error. A significant advantage is the ease of implementation.
A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane or hyperplane. We can use this operation to calculate the QR factorization of an m-by-n matrix with m ≥ n.
Q can be used to reflect a vector in such a way that all coordinates but one disappear.
Let be an arbitrary real m-dimensional column vector of such that for a scalar α. If the algorithm is implemented using floating-point arithmetic, then α should get the opposite sign as the k-th coordinate of , where is to be the pivot coordinate after which all entries are 0 in matrix A's final upper triangular form, to avoid loss of significance. In the complex case, set
and substitute transposition by conjugate transposition in the construction of Q below.
Then, where is the vector [1 0 ⋯ 0]T, ||·|| is the Euclidean norm and is an m×m identity matrix, set
Or, if is complex
is an m-by-m Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), and
This can be used to gradually transform an m-by-n matrix A to upper triangular form. First, we multiply A with the Householder matrix Q1 we obtain when we choose the first matrix column for x. This results in a matrix Q1A with zeros in the left column (except for the first row).
This can be repeated for A′ (obtained from Q1A by deleting the first row and first column), resulting in a Householder matrix Q′2. Note that Q′2 is smaller than Q1. Since we want it really to operate on Q1A instead of A′ we need to expand it to the upper left, filling in a 1, or in general:
The use of Householder transformations is inherently the most simple of the numerically stable QR decomposition algorithms due to the use of reflections as the mechanism for producing zeroes in the R matrix. However, the Householder reflection algorithm is bandwidth heavy and not parallelizable, as every reflection that produces a new zero element changes the entirety of both Q and R matrices.
QR decompositions can also be computed with a series of Givens rotations. Each rotation zeroes an element in the subdiagonal of the matrix, forming the R matrix. The concatenation of all the Givens rotations forms the orthogonal Q matrix.
In practice, Givens rotations are not actually performed by building a whole matrix and doing a matrix multiplication. A Givens rotation procedure is used instead which does the equivalent of the sparse Givens matrix multiplication, without the extra work of handling the sparse elements. The Givens rotation procedure is useful in situations where only relatively few off-diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations.
First, we need to form a rotation matrix that will zero the lowermost left element, . We form this matrix using the Givens rotation method, and call the matrix . We will first rotate the vector , to point along the X axis. This vector has an angle . We create the orthogonal Givens rotation matrix, :
And the result of now has a zero in the element.
We can similarly form Givens matrices and , which will zero the sub-diagonal elements and , forming a triangular matrix . The orthogonal matrix is formed from the product of all the Givens matrices . Thus, we have , and the QR decomposition is .
The QR decomposition via Givens rotations is the most involved to implement, as the ordering of the rows required to fully exploit the algorithm is not trivial to determine. However, it has a significant advantage in that each new zero element affects only the row with the element to be zeroed (i) and a row above (j). This makes the Givens rotation algorithm more bandwidth efficient and parallelizable than the Householder reflection technique.
Connection to a determinant or a product of eigenvalues
We can use QR decomposition to find the determinant of a square matrix. Suppose a matrix is decomposed as . Then we have
can be chosen such that . Thus,
where the are the entries on the diagonal of . Furthermore, because the determinant equals the product of the eigenvalues, we have
where the are eigenvalues of .
We can extend the above properties to a non-square complex matrix by introducing the definition of QR decomposition for non-square complex matrices and replacing eigenvalues with singular values.
Start with a QR decomposition for a non-square matrix A:
where denotes the zero matrix and is a unitary matrix.
From the properties of the SVD and the determinant of a matrix, we have
where the are the singular values of .
Note that the singular values of and are identical, although their complex eigenvalues may be different. However, if A is square, then
It follows that the QR decomposition can be used to efficiently calculate the product of the eigenvalues or singular values of a matrix.
Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step—column pivoting— and thus introduces a permutation matrixP:
Column pivoting is useful when A is (nearly) rank deficient, or is suspected of being so. It can also improve numerical accuracy. P is usually chosen so that the diagonal elements of R are non-increasing: . This can be used to find the (numerical) rank of A at lower computational cost than a singular value decomposition, forming the basis of so-called rank-revealing QR algorithms.
Using for solution to linear inverse problems
Compared to the direct matrix inverse, inverse solutions using QR decomposition are more numerically stable as evidenced by their reduced condition numbers.
To solve the underdetermined () linear problem where the matrix has dimensions and rank , first find the QR factorization of the transpose of :, where Q is an orthogonal matrix (i.e. ), and R has a special form: . Here is a square right triangular matrix, and the zero matrix has dimension . After some algebra, it can be shown that a solution to the inverse problem can be expressed as: where one may either find by Gaussian elimination or compute directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.
To find a solution to the overdetermined () problem which minimizes the norm , first find the QR factorization of :. The solution can then be expressed as , where is an matrix containing the first columns of the full orthonormal basis and where is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this without explicitly inverting . ( and are often provided by numerical libraries as an "economic" QR decomposition.)