QR decomposition

In linear algebra, a QR decomposition, also known as a QR factorization or QU factorization, is a decomposition of a matrix "A" into a product "A" = "QR" of an orthogonal matrix "Q" and an upper triangular matrix "R". QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.
Cases and definitions.
Square matrix.
Any real square matrix "A" may be decomposed as
where "Q" is an orthogonal matrix (its columns are orthogonal unit vectors meaning and "R" is an upper triangular matrix (also called right triangular matrix). If "A" is invertible, then the factorization is unique if we require the diagonal elements of "R" to be positive.
If instead "A" is a complex square matrix, then there is a decomposition "A" = "QR" where "Q" is a unitary matrix (so 
If "A" has "n" linearly 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 . The fact that any column "k" of "A" only depends on the first "k" columns of "Q" is responsible for the triangular form of "R".
Rectangular matrix.
More generally, we can factor a complex "m"×"n" matrix "A", with , as the product of an "m"×"m" unitary matrix "Q" 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 "R"1 is an "n"×"n" upper triangular matrix, 0 is an zero matrix, "Q"1 is "m"×"n", "Q"2 is , and "Q"1 and "Q"2 both have orthogonal columns.
 call "Q"1"R"1 the "thin QR factorization" of "A"; Trefethen and Bau call this the "reduced QR factorization". If "A" is of full rank "n" and we require that the diagonal elements of "R"1 are positive then "R"1 and "Q"1 are unique, but in general "Q"2 is not. "R"1 is then equal to the upper triangular factor of the Cholesky decomposition of "A" "A" (= "A"T"A" if "A" is real).
QL, RQ and LQ decompositions.
Analogously, we can define QL, RQ, and LQ decompositions, with "L" being a "lower" triangular matrix.
Computing the QR decomposition.
There are several methods for actually computing the QR decomposition, such as by means of the Gram–Schmidt process, Householder transformations, or Givens rotations. Each has a number of advantages and disadvantages.
Using the Gram–Schmidt process.
Consider the Gram–Schmidt process applied to the columns of the full column rank matrix with inner product formula_3 (or formula_4 for the complex case).
Define the projection:
then:
We can now express the formula_7s over our newly computed orthonormal basis:
where This can be written in matrix form:
where:
and
Example.
Consider the decomposition of
Recall that an orthonormal matrix formula_13 has the property 
Then, we can calculate formula_13 by means of Gram–Schmidt as follows:
Thus, we have
Relation to RQ decomposition.
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.
Advantages and disadvantages.
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 however is the ease of implementation, which makes this a useful algorithm to use for prototyping if a pre-built linear algebra library is unavailable.
Using Householder reflections.
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 formula_17 with .
"Q" can be used to reflect a vector in such a way that all coordinates but one disappear.
Let formula_18 be an arbitrary real "m"-dimensional column vector of formula_17 such that formula_20 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 formula_21 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 formula_23 is the vector [1 0 ⋯ 0]T, ||·|| is the Euclidean norm and formula_24 is an "m"×"m" identity matrix, set
Or, if formula_17 is complex
formula_13 is an "m"-by-"m" Householder matrix 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 "Q"1 we obtain when we choose the first matrix column for x. This results in a matrix "Q"1"A" with zeros in the left column (except for the first row).
This can be repeated for "A"′ (obtained from "Q"1"A" by deleting the first row and first column), resulting in a Householder matrix "Q"′2. Note that "Q"′2 is smaller than "Q"1. Since we want it really to operate on "Q"1"A" instead of "A"′ we need to expand it to the upper left, filling in a 1, or in general:
After formula_32 iterations of this process, 
is an upper triangular matrix. So, with
formula_9 is a QR decomposition of formula_17.
This method has greater numerical stability than the Gram–Schmidt method above.
The following table gives the number of operations in the "k"-th step of the QR-decomposition by the Householder transformation, assuming a square matrix with size "n".
Summing these numbers over the steps (for a square matrix of size "n"), the complexity of the algorithm (in terms of floating point multiplications) is given by
Example.
Let us calculate the decomposition of
First, we need to find a reflection that transforms the first column of matrix "A", vector into 
Now,
and
Here,
Therefore
Now observe:
so we already have almost a triangular matrix. We only need to zero the (3, 2) entry.
Take the (1, 1) minor, and then apply the process again to
By the same method as above, we obtain the matrix of the Householder transformation
after performing a direct sum with 1 to make sure the next step in the process works properly.
Now, we find
Or, to four decimal digits,
The matrix "Q" is orthogonal and "R" is upper triangular, so is the required QR decomposition.
Advantages and disadvantages.
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.
Using Givens rotations.
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 a relatively few off diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations.
Example.
Let us calculate the decomposition of
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 formula_51. We will first rotate the vector to point along the "X" axis. This vector has an angle We create the orthogonal Givens rotation matrix, formula_51:
And the result of formula_54 now has a zero in the formula_55 element.
We can similarly form Givens matrices formula_57 and which will zero the sub-diagonal elements formula_58 and forming a triangular matrix The orthogonal matrix formula_59 is formed from the product of all the Givens matrices Thus, we have and the "QR" decomposition is 
Advantages and disadvantages.
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 formula_60 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 absolute value of the determinant of a square matrix. Suppose a matrix is decomposed as formula_9. Then we have
Since "Q" is unitary, Thus,
where the formula_64 are the entries on the diagonal of "R".
Furthermore, because the determinant equals the product of the eigenvalues, we have
where the formula_66 are eigenvalues of 
We can extend the above properties to non-square complex matrix formula_17 by introducing the definition of QR decomposition for non-square complex matrix and replacing eigenvalues with singular values.
Suppose a QR decomposition for a non-square matrix "A":
where formula_69 denotes the zero matrix and formula_13 is a unitary matrix.
From the properties of SVD and determinant of matrix, we have
where the formula_72 are the singular values of 
Note that the singular values of formula_17 and formula_74 are identical, although their complex eigenvalues may be different. However, if "A" is square, the following is true:
In conclusion, QR decomposition can be used efficiently to calculate the product of the eigenvalues or singular values of a matrix.
Column pivoting.
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 matrix "P":
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: formula_77. 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 [Parker, Geophysical Inverse Theory, Ch1.13].
To solve the underdetermined linear problem formula_78 where the matrix formula_17 has dimensions formula_80 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: formula_81. Here formula_82 is a square formula_83 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: formula_84 where one may either find formula_85 by Gaussian elimination or compute formula_86 directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.
To find a solution formula_87 to the overdetermined problem formula_78 which minimizes the norm - \mathbf{b}\right\|</math>,}} first find the QR factorization of The solution can then be expressed as where formula_89 is an formula_80 matrix containing the first formula_91 columns of the full orthonormal basis formula_13 and where formula_82 is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this formula_94 without explicitly inverting (formula_89 and formula_82 are often provided by numerical libraries as an "economic" QR decomposition.)
Generalizations.
Iwasawa decomposition generalizes QR decomposition to semi-simple Lie groups.