QR factorization

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
: $A = QR,$

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'' corresponds to 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'':
:$A = QR = Q \begin R_1 \\ 0 \end = \begin Q_1 & Q_2 \end \begin R_1 \\ 0 \end = Q_1 R_1,$

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 $\langle\mathbf, \mathbf\rangle = \mathbf^\textsf \mathbf$ (or $\langle\mathbf, \mathbf\rangle = \mathbf^* \mathbf$ for the complex case).

Define the projection:
:$\operatorname_\mathbf = \frac$

then:
:$\begin \mathbf_1 &= \mathbf_1, & \mathbf_1 &= \frac \\ \mathbf_2 &= \mathbf_2 - \operatorname_ \mathbf_2, & \mathbf_2 &= \frac \\ \mathbf_3 &= \mathbf_3 - \operatorname_ \mathbf_3 - \operatorname_ \mathbf_3, & \mathbf_3 &= \frac \\ & \;\; \vdots & & \;\; \vdots \\ \mathbf_k &= \mathbf_k - \sum_^\operatorname_ \mathbf_k,& \mathbf_k &= \frac \end$

We can now express the $\mathbf_i$s over our newly computed orthonormal basis:
:$\begin \mathbf_1 &= \left\langle\mathbf_1, \mathbf_1\right\rangle \mathbf_1 \\ \mathbf_2 &= \left\langle\mathbf_1, \mathbf_2\right\rangle \mathbf_1 + \left\langle\mathbf_2, \mathbf_2\right\rangle \mathbf_2 \\ \mathbf_3 &= \left\langle\mathbf_1, \mathbf_3\right\rangle \mathbf_1 + \left\langle\mathbf_2, \mathbf_3\right\rangle \mathbf_2 + \left\langle\mathbf_3, \mathbf_3\right\rangle \mathbf_3 \\ &\;\;\vdots \\ \mathbf_k &= \sum_^k \left\langle \mathbf_j, \mathbf_k \right\rangle \mathbf_j \end$

where This can be written in matrix form:
:$A = QR$

where:
:$Q = \begin\mathbf_1 & \cdots & \mathbf_n\end$

and
:$R = \begin \langle\mathbf_1, \mathbf_1\rangle & \langle\mathbf_1, \mathbf_2\rangle & \langle\mathbf_1, \mathbf_3\rangle & \cdots & \langle\mathbf_1, \mathbf_n\rangle \\ 0 & \langle\mathbf_2, \mathbf_2\rangle & \langle\mathbf_2, \mathbf_3\rangle & \cdots & \langle\mathbf_2, \mathbf_n\rangle \\ 0 & 0 & \langle\mathbf_3, \mathbf_3\rangle & \cdots & \langle\mathbf_3, \mathbf_n\rangle \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \langle\mathbf_n, \mathbf_n\rangle \\ \end.$

Example

Consider the decomposition of
: $A = \begin 12 & -51 & 4 \\ 6 & 167 & -68 \\ -4 & 24 & -41 \end.$

Recall that an orthonormal matrix $Q$ has the property

Then, we can calculate $Q$ by means of Gram–Schmidt as follows:
: $\begin U = \begin \mathbf u_1 & \mathbf u_2 & \mathbf u_3 \end &= \begin 12 & -69 & -58/5 \\ 6 & 158 & 6/5 \\ -4 & 30 & -33 \end; \\ Q = \begin \frac & \frac & \frac \end &= \begin 6/7 & -69/175 & -58/175 \\ 3/7 & 158/175 & 6/175 \\ -2/7 & 6/35 & -33/35 \end. \end$

Thus, we have
: $\begin Q^\textsf A &= Q^\textsfQ\,R = R; \\ R &= Q^\textsfA = \begin 14 & 21 & -14 \\ 0 & 175 & -70 \\ 0 & 0 & 35 \end. \end$

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, which is almost never the case.

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 $A$ with .

''Q'' can be used to reflect a vector in such a way that all coordinates but one disappear.

Let $\mathbf$ be an arbitrary real ''m''-dimensional column vector of $A$ such that $\, \mathbf\, = , \alpha,$ 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 $x_k$ 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
:$\alpha = -e^ \, \mathbf\,$
and substitute transposition by conjugate transposition in the construction of ''Q'' below.

Then, where $\mathbf_1$ is the vector 0 ⋯ 0sup>T, , , ·, , is the Euclidean norm and $I$ is an ''m''×''m'' identity matrix, set
: $\begin \mathbf &= \mathbf - \alpha\mathbf_1, \\ \mathbf &= \frac, \\ Q &= I - 2 \mathbf\mathbf^\textsf. \end$

Or, if $A$ is complex
: $Q = I - 2\mathbf\mathbf^*.$

$Q$ is an ''m''-by-''m'' Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), and
: $Q\mathbf = \begin \alpha \\ 0 \\ \vdots \\ 0 \end.$

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).
: $Q_1A = \begin \alpha_1 & \star & \cdots & \star \\ 0 & & & \\ \vdots & & A' & \\ 0 & & & \end$

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:
:$Q_k = \begin I_ & 0 \\ 0 & Q_k' \end.$

After $t$ iterations of this process,
:$R = Q_t \cdots Q_2 Q_1 A$

is an upper triangular matrix. So, with
:$\begin Q^\textsf &= Q_t \cdots Q_2 Q_1, \\ Q &= Q_1^\textsf Q_2^\textsf \cdots Q_t^\textsf, \\ &= Q_1 Q_2 \cdots Q_t, \end$

$A = QR$ is a QR decomposition of $A$.

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
:$\fracn^3 + n^2 + \fracn - 2 = O\left(n^3\right).$

Example

Let us calculate the decomposition of
: $A = \begin 12 & -51 & 4 \\ 6 & 167 & -68 \\ -4 & 24 & -41 \end.$

First, we need to find a reflection that transforms the first column of matrix ''A'', vector into

Now,
: $\mathbf = \mathbf - \alpha\mathbf_1,$

and
: $\mathbf = \frac.$

Here,
: $\alpha = 14$ and $\mathbf = \mathbf_1 = \begin 12 & 6 & -4 \end^\textsf$

Therefore
: $\mathbf = \begin -2 & 6 & -4 \end^\textsf = 2 \begin -1 & 3 & -2 \end^\textsf$ and and then
: $\begin Q_1 = &I - \frac \begin -1 \\ 3 \\ -2 \end \begin -1 & 3 & -2 \end \\ = &I - \frac\begin 1 & -3 & 2 \\ -3 & 9 & -6 \\ 2 & -6 & 4 \end \\ = &\begin 6/7 & 3/7 & -2/7 \\ 3/7 & -2/7 & 6/7 \\ -2/7 & 6/7 & 3/7 \\ \end. \end$

Now observe:
:$Q_1A = \begin 14 & 21 & -14 \\ 0 & -49 & -14 \\ 0 & 168 & -77 \end,$

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
:$A' = M_ = \begin -49 & -14 \\ 168 & -77 \end.$

By the same method as above, we obtain the matrix of the Householder transformation
:$Q_2 = \begin 1 & 0 & 0 \\ 0 & -7/25 & 24/25 \\ 0 & 24/25 & 7/25 \end$

after performing a direct sum with 1 to make sure the next step in the process works properly.

Now, we find
:$Q = Q_1^\textsf Q_2^\textsf = \begin 6/7 & -69/175 & -58/175 \\ 3/7 & 158/175 & 6/175 \\ -2/7 & 6/35 & -33/35 \end.$

Or, to four decimal digits,
:$\begin Q &= Q_1^\textsf Q_2^\textsf = \begin 0.8571 & -0.3943 & -0.3314 \\ 0.4286 & 0.9029 & 0.0343 \\ -0.2857 & 0.1714 & -0.9429 \end \\ R &= Q_2 Q_1 A = Q^\textsf A = \begin 14 & 21 & -14 \\ 0 & 175 & -70 \\ 0 & 0 & -35 \end. \end$

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 relatively few off-diagonal elements need to be zeroed, and is more easily parallelized than Householder transformations.

Example

Let us calculate the decomposition of
: $A = \begin 12 & -51 & 4 \\ 6 & 167 & -68 \\ -4 & 24 & -41 \end.$

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 $G_1$. We will first rotate the vector to point along the ''X'' axis. This vector has an angle We create the orthogonal Givens rotation matrix, $G_1$:

:$\begin G_1 &= \begin \cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end \\ &\approx \begin 0.94868 & 0 & -0.31622 \\ 0 & 1 & 0 \\ 0.31622 & 0 & 0.94868 \end \end$

And the result of $G_1A$ now has a zero in the $a_$ element.
:$G_1A \approx \begin 12.64911 & -55.97231 & 16.76007 \\ 6 & 167 & -68 \\ 0 & 6.64078 & -37.6311 \end$

We can similarly form Givens matrices $G_2$ and which will zero the sub-diagonal elements $a_$ and forming a triangular matrix The orthogonal matrix $Q^\textsf$ 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 $a_$ 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 $A = QR$. Then we have
$\det A = \det Q \det R.$

$Q$ can be chosen such that $\det Q = 1$. Thus,
$\det A = \det R = \prod_i r_$

where the $r_$ are the entries on the diagonal of $R$. Furthermore, because the determinant equals the product of the eigenvalues, we have
$\prod_ r_ = \prod_ \lambda_$

where the $\lambda_i$ are eigenvalues of $A$.

We can extend the above properties to a non-square complex matrix $A$ 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'':
: $A = Q \begin R \\ 0 \end, \qquad Q^* Q = I$

where $0$ denotes the zero matrix and $Q$ is a unitary matrix.

From the properties of the SVD and the determinant of a matrix, we have
:$\Big, \prod_i r_\Big, = \prod_i\sigma_,$

where the $\sigma_i$ are the singular values of

Note that the singular values of $A$ and $R$ are identical, although their complex eigenvalues may be different. However, if ''A'' is square, then
:$= \Big, \prod_i \lambda_i\Big, .$

It follows that the QR decomposition can be used to efficiently 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'':
:$AP = QR\quad \iff\quad A = QRP^\textsf$

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: $\left, r_\ \ge \left, r_\ \ge \cdots \ge \left, r_\$. 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 arker, Geophysical Inverse Theory, Ch1.13

To solve the underdetermined linear problem $A \mathbf x = \mathbf b$ where the matrix $A$ has dimensions $m \times n$ 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: R = \left begin R_1 \\ 0 \end\right/math>. Here $R_1$ is a square $m \times m$ 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: \mathbf x = Q \left begin
\left(R_1^\textsf\right)^ \mathbf b \\
0
\end\right/math> where one may either find $R_1^$ by Gaussian elimination or compute $\left(R_1^\textsf\right)^ \mathbf b$ directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.

To find a solution $\hat$ to the overdetermined problem $A \mathbf x = \mathbf b$ which minimizes the norm first find the QR factorization of The solution can then be expressed as where $Q_1$ is an $m \times n$ matrix containing the first $n$ columns of the full orthonormal basis $Q$ and where $R_1$ is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this $\hat$ without explicitly inverting ($Q_1$ and $R_1$ are often provided by numerical libraries as an "economic" QR decomposition.)

Generalizations

Iwasawa decomposition generalizes QR decomposition to semi-simple Lie groups.

See also

* Polar decomposition
* Eigenvalue decomposition
* Spectral decomposition
* LU decomposition
* Singular value decomposition

References

Further reading

* .
*
*

External links

Online Matrix Calculator
Performs QR decomposition of matrices.

gives details of subroutines to calculate the QR decomposition
Mathematica users manual
gives details and examples of routines to calculate QR decomposition
ALGLIB
includes a partial port of the LAPACK to C++, C#, Delphi, etc.

Includes C++ implementation of QR decomposition.

{{Numerical linear algebra

Matrix decompositions
Numerical linear algebra