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 orthonormal matrix Q and an upper triangular matrix R. QR decomposition is often used to solve the linear least squares (LLS) 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
: <math> A = QR, </math>
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 the conjugate transpose
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.
:<math>\alpha = -e^{i \arg x_k} \|\mathbf{x}\|</math>
and substitute transposition by conjugate transposition in the construction of Q below.
Then, where <math>\mathbf{e}_1</math> is the vector , is the Euclidean norm and <math>I</math> is an identity matrix, set
: <math>\begin{align}
\mathbf{u} &= \mathbf{x} - \alpha\mathbf{e}_1, \\
\mathbf{v} &= \frac{\mathbf{u{\|\mathbf{u}\|}, \\
Q &= I - 2 \mathbf{v}\mathbf{v}^\textsf{T}.
\end{align}</math>
Or, if <math>A</math> is complex
: <math>Q = I - 2\mathbf{v}\mathbf{v}^\dagger.</math>
<math>Q</math> is an m-by-m Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), and
: <math>Q\mathbf{x} = \begin{bmatrix} \alpha \\ 0 \\ \vdots \\ 0 \end{bmatrix}.</math>
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<sub>1</sub> we obtain when we choose the first matrix column for x. This results in a matrix Q<sub>1</sub>A with zeros in the left column (except for the first row).
: <math>Q_1A = \begin{bmatrix}
\alpha_1 & \star & \cdots & \star \\
0 & & & \\
\vdots & & A' & \\
0 & & &
\end{bmatrix}</math>
This can be repeated for A′ (obtained from Q<sub>1</sub>A by deleting the first row and first column), resulting in a Householder matrix Q′<sub>2</sub>. Note that Q′<sub>2</sub> is smaller than Q<sub>1</sub>. Since we want it really to operate on Q<sub>1</sub>A instead of A′ we need to expand it to the upper left, filling in a 1, or in general:
:<math>Q_k = \begin{bmatrix}
I_{k-1} & 0 \\
0 & Q_k'
\end{bmatrix}.</math>
After <math>t</math> iterations of this process,
:<math>R = Q_t \cdots Q_2 Q_1 A</math>
is an upper triangular matrix. So, with
:<math>\begin{align}
Q^\textsf{T} &= Q_t \cdots Q_2 Q_1, \\
Q &= Q_1^\textsf{T} Q_2^\textsf{T} \cdots Q_t^\textsf{T}
\end{align}</math>
<math>A = QR</math> is a QR decomposition of <math>A</math>.
This method has greater numerical stability than the Gram–Schmidt method above.<!--See the below example, and compare above-->
In numerical tests the computed factors <math>Q_c</math> and <math>R_c</math> satisfy
<math>\frac{\|Q R - Q_c R_c\|_\infty}{\|A\|_\infty} = O(\varepsilon)</math>
at machine precision. Also, orthogonality is preserved: <math>\|Q_c^\mathsf{T} Q_c - I\|_\infty = O(\varepsilon)</math>. However, the accuracy of <math>Q_c</math> and <math>R_c</math> decrease with condition number:
<math>\|Q - Q_c\|_\infty = O(\varepsilon\,\kappa_\infty(A)),\quad
\frac{\|R - R_c\|_\infty}{\|R\|_\infty} = O(\varepsilon\,\kappa_\infty(A)).</math>
For a well-conditioned example (<math>n=4000</math>, <math>\kappa_\infty(A)\approx3\times10^{3}</math>):
<math>\frac{\|Q R - Q_c R_c\|_\infty}{\|A\|_\infty} \approx 1.6\times10^{-15},</math>
<math>\|Q - Q_c\|_\infty \approx 1.6\times10^{-15},</math>
<math>\frac{\|R - R_c\|_\infty}{\|R\|_\infty} \approx 4.3\times10^{-14},</math>
<math>\|Q_c^\mathsf{T}Q_c - I\|_\infty \approx 1.1\times10^{-13}.</math>
In an ill-conditioned test (<math>n=4000</math>, <math>\kappa_\infty(A)\approx4\times10^{18}</math>):
<math>\frac{\|Q R - Q_c R_c\|_\infty}{\|A\|_\infty} \approx 1.3\times10^{-15},</math>
<math>\|Q - Q_c\|_\infty \approx 5.2\times10^{-4},</math>
<math>\frac{\|R - R_c\|_\infty}{\|R\|_\infty} \approx 1.2\times10^{-4},</math>
<math>\|Q_c^\mathsf{T}Q_c - I\|_\infty \approx 1.1\times10^{-13}.</math>
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.
{| class="wikitable"
|-
! Operation
! Number of operations in the k-th step
|-
| Multiplications
| <math>2(n - k + 1)^2</math>
|-
| Additions
| <math>(n - k + 1)^2 + (n - k + 1)(n - k) + 2 </math>
|-
| Division
| <math>1</math>
|-
| Square root
| <math>1</math>
|}
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
:<math>\frac{2}{3}n^3 + n^2 + \frac{1}{3}n - 2 = O\left(n^3\right).</math>
Example
Let us calculate the decomposition of
: <math>A = \begin{bmatrix}
12 & -51 & 4 \\
6 & 167 & -68 \\
-4 & 24 & -41
\end{bmatrix}.</math>
First, we need to find a reflection that transforms the first column of matrix A, vector into
Now,
: <math>\mathbf{u} = \mathbf{x} - \alpha\mathbf{e}_1,</math>
and
: <math>\mathbf{v} = \frac{\mathbf{u{\|\mathbf{u}\|}.</math>
Here,
: <math>\alpha = 14</math> and <math>\mathbf{x} = \mathbf{a}_1 = \begin{bmatrix} 12 & 6 & -4 \end{bmatrix}^\textsf{T}</math>
Therefore
: <math>\mathbf{u} = \begin{bmatrix} -2 & 6 & -4 \end{bmatrix}^\textsf{T} = 2 \begin{bmatrix} -1 & 3 & -2 \end{bmatrix}^\textsf{T}</math> and \begin{bmatrix} -1 & 3 & -2 \end{bmatrix}^\textsf{T}</math>, and then
: <math>\begin{align}
Q_1
={} &I - \frac{2}{\sqrt{14}\sqrt{14
\begin{bmatrix} -1 \\ 3 \\ -2 \end{bmatrix}
\begin{bmatrix} -1 & 3 & -2 \end{bmatrix} \\
={} &I - \frac{1}{7}\begin{bmatrix}
1 & -3 & 2 \\
-3 & 9 & -6 \\
2 & -6 & 4
\end{bmatrix} \\
={} &\begin{bmatrix}
6/7 & 3/7 & -2/7 \\
3/7 & -2/7 & 6/7 \\
-2/7 & 6/7 & 3/7 \\
\end{bmatrix}.
\end{align}</math>
Now observe:
:<math>Q_1A = \begin{bmatrix}
14 & 21 & -14 \\
0 & -49 & -14 \\
0 & 168 & -77
\end{bmatrix},</math>
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
:<math>A' = M_{11} = \begin{bmatrix}
-49 & -14 \\
168 & -77
\end{bmatrix}.</math>
By the same method as above, we obtain the matrix of the Householder transformation
:<math>Q_2 = \begin{bmatrix}
1 & 0 & 0 \\
0 & -7/25 & 24/25 \\
0 & 24/25 & 7/25
\end{bmatrix}</math>
after performing a direct sum with 1 to make sure the next step in the process works properly.
Now, we find
:<math>Q = Q_1^\textsf{T} Q_2^\textsf{T} = \begin{bmatrix}
6/7 & -69/175 & 58/175 \\
3/7 & 158/175 & -6/175 \\
-2/7 & 6/35 & 33/35
\end{bmatrix}. </math>
Or, to four decimal digits,
:<math>\begin{align}
Q &= Q_1^\textsf{T} Q_2^\textsf{T} = \begin{bmatrix}
0.8571 & -0.3943 & 0.3314 \\
0.4286 & 0.9029 & -0.0343 \\
-0.2857 & 0.1714 & 0.9429
\end{bmatrix} \\
R &= Q_2 Q_1 A = Q^\textsf{T} A = \begin{bmatrix}
14 & 21 & -14 \\
0 & 175 & -70 \\
0 & 0 & -35
\end{bmatrix}.
\end{align}</math>
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 difficult to parallelize, as every reflection that produces a new zero element changes the entirety of both Q and R matrices.
Parallel implementation of Householder QR
The Householder QR method can be implemented in parallel with algorithms such as the TSQR algorithm (which stands for Tall Skinny QR). This algorithm can be applied in the case when the matrix A has m >> n.
This algorithm uses a binary reduction tree to compute local householder QR decomposition at each node in the forward pass, and re-constitute the Q matrix in the backward pass. The binary tree structure aims at decreasing the amount of communication between processor to increase performance.
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
: <math>A = \begin{bmatrix}
12 & -51 & 4 \\
6 & 167 & -68 \\
-4 & 24 & -41
\end{bmatrix}.</math>
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 <math>G_1</math>. We will first rotate the vector to point along the X axis. This vector has an angle We create the orthogonal Givens rotation matrix, <math>G_1</math>:
:<math>\begin{align}
G_1 &= \begin{bmatrix}
\cos(\theta) & 0 & -\sin(\theta) \\
0 & 1 & 0 \\
\sin(\theta) & 0 & \cos(\theta)
\end{bmatrix} \\
&\approx \begin{bmatrix}
0.94868 & 0 & -0.31622 \\
0 & 1 & 0 \\
0.31622 & 0 & 0.94868
\end{bmatrix}
\end{align}</math>
And the result of <math>G_1A</math> now has a zero in the <math>a_{31}</math> element.
:<math>G_1A \approx \begin{bmatrix}
12.64911 & -55.97231 & 16.76007 \\
6 & 167 & -68 \\
0 & 6.64078 & -37.6311
\end{bmatrix}</math>
We can similarly form Givens matrices <math>G_2</math> and which will zero the sub-diagonal elements <math>a_{21}</math> and forming a triangular matrix The orthogonal matrix <math>Q^\textsf{T}</math> 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 <math>a_{ij}</math> 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.
Using fast matrix multiplication
It is possible to compute the QR decomposition in a fast way with the use of fast matrix multiplication algorithms in the time <math>O({n^\omega })</math> for <math>~2.37 \le \omega < 3</math>.
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 <math>A = QR</math>. Then we have
<math display='block'>\det A = \det Q \det R.</math>
<math>Q</math> can be chosen such that <math>\det Q = 1</math>. Thus,
<math display='block'>\det A = \det R = \prod_i r_{ii}</math>
where the <math>r_{ii}</math> are the entries on the diagonal of <math>R</math>. Furthermore, because the determinant equals the product of the eigenvalues, we have
<math display='block'>\prod_{i} r_{ii} = \prod_{i} \lambda_{i}</math>
where the <math>\lambda_i</math> are eigenvalues of <math>A</math>.
We can extend the above properties to a non-square complex matrix <math>A</math> 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:
: <math>A = Q \begin{bmatrix} R \\ 0 \end{bmatrix}, \qquad Q^\dagger Q = I</math>
where <math>0</math> denotes the zero matrix and <math>Q</math> is a unitary matrix.
From the properties of the singular value decomposition (SVD) and the determinant of a matrix, we have
:<math>\Big|\prod_i r_{ii}\Big| = \prod_i\sigma_{i},</math>
where the <math>\sigma_i</math> are the singular values of
Note that the singular values of <math>A</math> and <math>R</math> are identical, although their complex eigenvalues may be different. However, if A is square, then
:<math>{\prod_i \sigma_i} = \Big|\prod_i \lambda_i\Big|.</math>
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:
:<math>AP = QR\quad \iff\quad A = QRP^\textsf{T}</math>
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: <math>\left|r_{11}\right| \ge \left|r_{22}\right| \ge \cdots \ge \left|r_{nn}\right|</math>. 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 <math>A \mathbf x = \mathbf b</math> where the matrix <math>A</math> has dimensions <math>m \times n</math> 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: <math>R = \left[\begin{smallmatrix} R_1 \\ 0 \end{smallmatrix}\right]</math>. Here <math>R_1</math> is a square <math>m \times m</math> 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: <math>\mathbf x = Q \left[\begin{smallmatrix}
\left(R_1^\textsf{T}\right)^{-1} \mathbf b \\
0
\end{smallmatrix}\right]</math> where one may either find <math>R_1^{-1}</math> by Gaussian elimination or compute <math>\left(R_1^\textsf{T}\right)^{-1} \mathbf b</math> directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.
To find a solution <math>\hat{\mathbf x}</math> to the overdetermined problem <math>A \mathbf x = \mathbf b</math> which minimizes the norm - \mathbf{b}\right\|</math>, first find the QR factorization of The solution can then be expressed as where <math>Q_1</math> is an <math>m \times n</math> matrix containing the first <math>n</math> columns of the full orthonormal basis <math>Q</math> and where <math>R_1</math> is as before. Equivalent to the underdetermined case, back substitution can be used to quickly and accurately find this <math>\hat{\mathbf{x</math> without explicitly inverting (<math>Q_1</math> and <math>R_1</math> 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
- Eigendecomposition (spectral decomposition)
- LU decomposition
- Singular value decomposition