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 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.

Any real square matrix A may be decomposed as

\(A = QR,\)

where Q is an orthogonal matrix (its columns are orthogonal unit vectors meaning \(Q^\textsf{T} = Q^{-1}\)) 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 \(Q^\dagger = Q^{-1}\)).

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 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×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:

\(A = QR = Q \begin{bmatrix} R_1 \\ 0 \end{bmatrix} = \begin{bmatrix} Q_1 & Q_2 \end{bmatrix} \begin{bmatrix} R_1 \\ 0 \end{bmatrix} = Q_1 R_1,\)

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 rank n 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 (= AA if A is real).

There are several methods for actually computing the QR decomposition, such as the Gram-Schmidt process, Householder transformations, or Givens rotations. Each has a number of advantages and disadvantages.

Consider the Gram-Schmidt process applied to the columns of the full column rank matrix \(A = \begin{bmatrix}\mathbf{a}_1 & \cdots & \mathbf{a}_n\end{bmatrix}\), with inner product \(\langle\mathbf{v}, \mathbf{w}\rangle = \mathbf{v}^\textsf{T} \mathbf{w}\) (or \(\langle\mathbf{v}, \mathbf{w}\rangle = \mathbf{v}^\dagger \mathbf{w}\) for the complex case).

Define the projection:

\(\operatorname{proj}_{\mathbf{u}}\mathbf{a} = \frac{\left\langle\mathbf{u}, \mathbf{a}\right\rangle}{\left\langle\mathbf{u}, \mathbf{u}\right\rangle}{\mathbf{u}}\)

then:

\(\begin{align} \mathbf{u}_1 &= \mathbf{a}_1, & \mathbf{e}_1 &= \frac{\mathbf{u}_1}{\|\mathbf{u}_1\|} \\ \mathbf{u}_2 &= \mathbf{a}_2 - \operatorname{proj}_{\mathbf{u}_1} \mathbf{a}_2, & \mathbf{e}_2 &= \frac{\mathbf{u}_2}{\|\mathbf{u}_2\|} \\ \mathbf{u}_3 &= \mathbf{a}_3 - \operatorname{proj}_{\mathbf{u}_1} \mathbf{a}_3 - \operatorname{proj}_{\mathbf{u}_2} \mathbf{a}_3, & \mathbf{e}_3 &= \frac{\mathbf{u}_3}{\|\mathbf{u}_3\|} \\ & \;\; \vdots & & \;\; \vdots \\ \mathbf{u}_k &= \mathbf{a}_k - \sum_{j=1}^{k-1}\operatorname{proj}_{\mathbf{u}_j} \mathbf{a}_k,& \mathbf{e}_k &= \frac{\mathbf{u}_k}{\|\mathbf{u}_k\|} \end{align}\)

We can now express the \(\mathbf{a}_i\)s over our newly computed orthonormal basis:

\(\begin{align} \mathbf{a}_1 &= \left\langle\mathbf{e}_1, \mathbf{a}_1\right\rangle \mathbf{e}_1 \\ \mathbf{a}_2 &= \left\langle\mathbf{e}_1, \mathbf{a}_2\right\rangle \mathbf{e}_1 + \left\langle\mathbf{e}_2, \mathbf{a}_2\right\rangle \mathbf{e}_2 \\ \mathbf{a}_3 &= \left\langle\mathbf{e}_1, \mathbf{a}_3\right\rangle \mathbf{e}_1 + \left\langle\mathbf{e}_2, \mathbf{a}_3\right\rangle \mathbf{e}_2 + \left\langle\mathbf{e}_3, \mathbf{a}_3\right\rangle \mathbf{e}_3 \\ &\;\;\vdots \\ \mathbf{a}_k &= \sum_{j=1}^k \left\langle \mathbf{e}_j, \mathbf{a}_k \right\rangle \mathbf{e}_j \end{align}\)

where \(\left\langle\mathbf{e}_i, \mathbf{a}_i\right\rangle = \left\|\mathbf{u}_i\right\|\). This can be written in matrix form:

\(A = QR\)

where:

\(Q = \begin{bmatrix}\mathbf{e}_1 & \cdots & \mathbf{e}_n\end{bmatrix}\)

and

Condensed: the full section is in Wikipedia.

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 m ≥ n.

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

Let \(\mathbf{x}\) be an arbitrary real m-dimensional column vector of \(A\) such that \(\|\mathbf{x}\| = |\alpha|\) for a scalar α. α should get the same sign as the \(k\)-th coordinate of \(\mathbf{x}\), where \(x_k\) is to be the pivot coordinate after which all entries are 0 in matrix A's final upper triangular form. If the algorithm is implemented using floating-point arithmetic, then α should get the opposite sign to avoid loss of significance (for example, when \(\mathbf{x}\) is almost collinear with \(\mathbf{e}_1\), \(\|\mathbf{u}\|\) becomes "small" and \(\mathbf{u} / \|\mathbf{u}\|\) is numerically unstable; the extreme case is \(\|\mathbf{u}\| = 0\), which causes the previous division to result in NaN).

In the complex case, set

\(\alpha = -e^{i \arg x_k} \|\mathbf{x}\|\)

and substitute transposition by conjugate transposition in the construction of Q below.

Then, where \(\mathbf{e}_1\) is the vector [1 0 ⋯ 0], || · || is the Euclidean norm and \(I\) is an m×m identity matrix, set

\(\begin{align} \mathbf{u} &= \mathbf{x} - \alpha\mathbf{e}_1, \\ \mathbf{v} &= \frac{\mathbf{u}}{\|\mathbf{u}\|}, \\ Q &= I - 2 \mathbf{v}\mathbf{v}^\textsf{T}. \end{align}\)

Or, if \(A\) is complex

\(Q = I - 2\mathbf{v}\mathbf{v}^\dagger.\)

\(Q\mathbf{x} = \begin{bmatrix} \alpha \\ 0 \\ \vdots \\ 0 \end{bmatrix}.\)

\(Q_1A = \begin{bmatrix} \alpha_1 & \star & \cdots & \star \\ 0 & & & \\ \vdots & & A' & \\ 0 & & & \end{bmatrix}\)

\(Q_k = \begin{bmatrix} I_{k-1} & 0 \\ 0 & Q_k' \end{bmatrix}.\)

\(R = Q_t \cdots Q_2 Q_1 A\)

\(\begin{align} Q^\textsf{T} &= Q_t \cdots Q_2 Q_1, \\ Q &= Q_1^\textsf{T} Q_2^\textsf{T} \cdots Q_t^\textsf{T} \end{align}\)

\(\frac{2}{3}n^3 + n^2 + \frac{1}{3}n - 2 = O\left(n^3\right).\)

Condensed: the full section is in Wikipedia.

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.

It is possible to compute the QR decomposition in a fast way with the use of fast matrix multiplication algorithms in the time \(O({n^\omega })\) for \(~2.37 \le \omega < 3\).

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, \[\det A = \det R = \prod_i r_{ii}\]

where the \(r_{ii}\) are the entries on the diagonal of \(R\). Furthermore, because the determinant equals the product of the eigenvalues, we have \[\prod_{i} r_{ii} = \prod_{i} \lambda_{i}\]

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:

\(A = Q \begin{bmatrix} R \\ 0 \end{bmatrix}, \qquad Q^\dagger Q = I\)

where \(0\) denotes the zero matrix and \(Q\) is a unitary matrix.

\(\Big|\prod_i r_{ii}\Big| = \prod_i\sigma_{i},\)

\({\prod_i \sigma_i} = \Big|\prod_i \lambda_i\Big|.\)

Condensed: the full section is in Wikipedia.

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:

\(AP = QR\quad \iff\quad A = QRP^\textsf{T}\)

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: \(\left|r_{11}\right| \ge \left|r_{22}\right| \ge \cdots \ge \left|r_{nn}\right|\). 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.

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 (\(m < n\)) linear problem \(A \mathbf x = \mathbf b\) where the matrix \(A\) has dimensions \(m \times n\) and rank \(m\), first find the QR factorization of the transpose of \(A\): \(A^\textsf{T} = QR\), where Q is an orthogonal matrix (i.e. \(Q^\textsf{T} = Q^{-1}\)), and R has a special form: \(R = \left[\begin{smallmatrix} R_1 \\ 0 \end{smallmatrix}\right]\). Here \(R_1\) is a square \(m \times m\) right triangular matrix, and the zero matrix has dimension \((n-m) \times m\). After some algebra, it can be shown that a solution to the inverse problem can be expressed as: \(\mathbf x = Q \left[\begin{smallmatrix} \left(R_1^\textsf{T}\right)^{-1} \mathbf b \\ 0 \end{smallmatrix}\right]\) where one may either find \(R_1^{-1}\) by Gaussian elimination or compute \(\left(R_1^\textsf{T}\right)^{-1} \mathbf b\) directly by forward substitution. The latter technique enjoys greater numerical accuracy and lower computations.

To find a solution \(\hat{\mathbf x}\) to the overdetermined (\(m \geq n\)) problem \(A \mathbf x = \mathbf b\) which minimizes the norm \(\left\|A \hat{\mathbf{x}} - \mathbf{b}\right\|\), first find the QR factorization of \(A\): \(A = QR\). The solution can then be expressed as \(\hat{\mathbf x} = R_1^{-1} \left(Q_1^\textsf{T} \mathbf{b}\right)\), 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 \(\hat{\mathbf{x}}\) without explicitly inverting \(R_1\). (\(Q_1\) and \(R_1\) are often provided by numerical libraries as an "economic" QR decomposition.)