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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.

**Linear algebra** is the branch of mathematics concerning linear equations such as

In the mathematical discipline of linear algebra, a **matrix decomposition** or **matrix factorization** is a factorization of a matrix into a product of matrices. There are many different matrix decompositions; each finds use among a particular class of problems.

In mathematics, a **matrix** is a rectangular *array* of numbers, symbols, or expressions, arranged in *rows* and *columns*. For example, the dimension of the matrix below is 2 × 3, because there are two rows and three columns:

- Cases and definitions
- Square matrix
- Rectangular matrix
- QL, RQ and LQ decompositions
- Computing the QR decomposition
- Using the Gram–Schmidt process
- Using Householder reflections
- Using Givens rotations
- Connection to a determinant or a product of eigenvalues
- Column pivoting
- Using for solution to linear inverse problems
- Generalizations
- See also
- References
- Further reading
- External links

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, hence the name). If *A* is invertible, then the factorization is unique if we require the diagonal elements of *R* to be positive.

An **orthogonal matrix** is a square matrix whose columns and rows are orthogonal unit vectors, i.e.

In mathematics, a **unit vector** in a normed vector space is a vector of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat": . The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as **d**. Two 2D direction vectors, **d1** and **d2** are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.

In the mathematical discipline of linear algebra, a **triangular matrix** is a special kind of square matrix. A square matrix is called **lower triangular** if all the entries *above* the main diagonal are zero. Similarly, a square matrix is called **upper triangular** if all the entries *below* the main diagonal are zero. A triangular matrix is one that is either lower triangular or upper triangular. A matrix that is both upper and lower triangular is called a diagonal matrix.

If instead *A* is a complex square matrix, then there is a decomposition *A* = *QR* where *Q* is a unitary matrix (so ).

In mathematics, a complex square matrix *U* is **unitary** if its conjugate transpose *U*^{∗} is also its inverse—that is, if

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*.^{ [1] } 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*.^{ [1] }

In mathematics, particularly linear algebra, an **orthonormal basis** for an inner product space *V* with finite dimension is a basis for *V* whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space **R**^{n} is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection is also orthonormal, and every orthonormal basis for **R**^{n} arises in this fashion.

In linear algebra, the **linear span** of a set S of vectors in a vector space is the smallest linear subspace that contains the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules.

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*:

where *R*_{1} is an *n*×*n* upper triangular matrix, *0* is an (*m* − *n*)×*n* zero matrix, *Q*_{1} is *m*×*n*, *Q*_{2} is *m*×(*m* − *n*), and *Q*_{1} and *Q*_{2} both have orthogonal columns.

Golub & Van Loan (1996 , §5.2) call *Q*_{1}*R*_{1} the *thin QR factorization* of *A*; Trefethen and Bau call this the *reduced QR factorization*.^{ [1] } 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).

In linear algebra, the **Cholesky decomposition** or **Cholesky factorization** is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations. It was discovered by André-Louis Cholesky for real matrices. When it is applicable, the Cholesky decomposition is roughly twice as efficient as the LU decomposition for solving systems of linear equations.....

Analogously, we can define QL, RQ, and LQ decompositions, with *L* being a *lower* triangular matrix.

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.

Consider the Gram–Schmidt process applied to the columns of the full column rank matrix , with inner product (or for the complex case).

Define the projection:

then:

We can now express the s over our newly computed orthonormal basis:

where . This can be written in matrix form:

where:

and

Consider the decomposition of

Recall that an orthonormal matrix has the property

Then, we can calculate by means of Gram–Schmidt as follows:

Thus, we have

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 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.

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

( Stoer & Bulirsch 2002 , p. 225) 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*-by-*m* identity matrix, set

Or, if is complex

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 iterations of this process, ,

is an upper triangular matrix. So, with

is a QR decomposition of .

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*.

Operation | Number of operations in the k-th step |
---|---|

Multiplications | |

Additions | |

Division | |

Square root |

Summing these numbers over the *n* − 1 steps (for a square matrix of size *n*), the complexity of the algorithm (in terms of floating point multiplications) is given by

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,

- and

Therefore

- and , and then

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 *A* = *QR* is the required QR-decomposition.

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

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 . 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 parallelisable than the Householder reflection technique.

We can use QR decomposition to find the absolute value of the determinant of a square matrix. Suppose a matrix is decomposed as . Then we have

Since *Q* is unitary, . Thus,

where are the entries on the diagonal of *R*.

Furthermore, because the determinant equals the product of the eigenvalues, we have

where are eigenvalues of .

We can extend the above properties to non-square complex matrix 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 is a zero matrix and is a unitary matrix.

From the properties of SVD and determinant of matrix, we have

where are singular values of .

Note that the singular values of and 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.

Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step -column pivoting- ^{ [2] } 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: . 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 [Parker, Geophysical Inverse Theory, Ch1.13].

To solve the underdetermined () linear problem where the matrix A has dimensions and rank , first find the QR factorization of the transpose of A: , 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 A: . 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.)

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

In quantum mechanics, **bra–ket notation** is a common notation for quantum states i.e. vectors in a complex Hilbert space on which an algebra of observables act. More generally the notation uses the angle brackets and a vertical bar, for a **ket** like to denote a vector in an abstract vector space and a **bra**, like to denote a linear functional on , i.e. a co-vector, an element of the dual vector space . The natural pairing of a linear functional with a vector is then written as . On Hilbert spaces, the scalar product gives an (anti-linear) identification of a vector ket with a linear functional bra . Using this notation, the scalar product . For the vector space , kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and operators are interpreted using matrix multiplication. If has the standard hermitian inner product , under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the hermitian conjugate .

In linear algebra, the **determinant** is a scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix. The determinant of a matrix *A* is denoted det(*A*), det *A*, or |*A*|. Geometrically, it can be viewed as the volume scaling factor of the linear transformation described by the matrix. This is also the signed volume of the *n*-dimensional parallelepiped spanned by the column or row vectors of the matrix. The determinant is positive or negative according to whether the linear mapping preserves or reverses the orientation of *n*-space.

In linear algebra, a symmetric real matrix is said to be **positive definite** if the scalar is strictly positive for every non-zero column vector of real numbers. Here denotes the transpose of . When interpreting as the output of an operator, , that is acting on an input, , the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes.

In mathematics, particularly linear algebra and numerical analysis, the **Gram–Schmidt process** is a method for orthonormalising a set of vectors in an inner product space, most commonly the Euclidean space **R**^{n} equipped with the standard inner product. The Gram–Schmidt process takes a finite, linearly independent set *S* = {*v*_{1}, ..., *v*_{k}} for *k* ≤ *n* and generates an orthogonal set *S′* = {*u*_{1}, ..., *u*_{k}} that spans the same *k*-dimensional subspace of **R**^{n} as *S*.

In linear algebra, a **symmetric matrix** is a square matrix that is equal to its transpose. Formally,

In linear algebra, the **singular value decomposition** (**SVD**) is a factorization of a real or complex matrix. It is the generalization of the eigendecomposition of a positive semidefinite normal matrix to any matrix via an extension of the polar decomposition. It has many useful applications in signal processing and statistics.

In mathematics, particularly in linear algebra, a **skew-symmetric****matrix** is a square matrix whose transpose equals its negative. That is, it satisfies the condition

In mathematics, and in particular linear algebra, a **pseudoinverse***A*^{+} of a matrix *A* is a generalization of the inverse matrix. The most widely known type of matrix pseudoinverse is the **Moore–Penrose inverse**, which was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse.

In linear algebra, a **Householder transformation** is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformation was introduced in 1958 by Alston Scott Householder.

In linear algebra and functional analysis, a **projection** is a linear transformation from a vector space to itself such that . That is, whenever is applied twice to any value, it gives the same result as if it were applied once (idempotent). It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object.

In physics, the **S-matrix** or **scattering matrix** relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).

In linear algebra, an **eigenvector** or **characteristic vector** of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it.

In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born–Oppenheimer approximation. This corresponds to a representation of the molecular wave function where the variables corresponding to the molecular geometry and the electronic degrees of freedom are separated. The non separable terms are due to the nuclear kinetic energy terms in the molecular Hamiltonian and are said to couple the potential energy surfaces. In the neighbourhood of an avoided crossing or conical intersection, these terms cannot be neglected. One therefore usually performs one unitary transformation from the adiabatic representation to the so-called **diabatic representation** in which the nuclear kinetic energy operator is diagonal. In this representation, the coupling is due to the electronic energy and is a scalar quantity which is significantly easier to estimate numerically.

The name **paravector** is used for the sum of a scalar and a vector in any Clifford algebra

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

**Numerical methods for linear least squares** entails the numerical analysis of linear least squares problems.

In pure and applied mathematics, particularly quantum mechanics and computer graphics and their applications, a **spherical basis** is the basis used to express **spherical tensors**. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions.

In pure and applied mathematics, quantum mechanics and computer graphics, a **tensor operator** generalizes the notion of operators which are scalars and vectors. A special class of these are **spherical tensor operators** which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a **representation operator**.

In mathematics, the **Frobenius inner product** is a binary operation that takes two matrices and returns a number. It is often denoted . The operation is a component-wise inner product of two matrices as though they are vectors. The two matrices must have the same dimension—same number of rows and columns—but are not restricted to be square matrices.

- Golub, Gene H.; Van Loan, Charles F. (1996),
*Matrix Computations*(3rd ed.), Johns Hopkins, ISBN 978-0-8018-5414-9 . - Horn, Roger A.; Johnson, Charles R. (1985),
*Matrix Analysis*, Cambridge University Press, ISBN 0-521-38632-2 . Section 2.8. - Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 2.10. QR Decomposition",
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8 - Stoer, Josef; Bulirsch, Roland (2002),
*Introduction to Numerical Analysis*(3rd ed.), Springer, ISBN 0-387-95452-X .

- Online Matrix Calculator Performs QR decomposition of matrices.
- LAPACK users manual 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.
- Eigen::QR Includes C++ implementation of QR decomposition.

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