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Linear algebra

Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...

, a QR decomposition (also called a QR factorization) of a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

is a decomposition

Matrix decomposition

In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. There are many different matrix decompositions; each finds use among a particular class of problems.- Example :...

of a matrix A into a product A=QR of an orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

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

Eigenvalue algorithm

In linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.-Characteristic polynomial:...

, the QR algorithm

QR algorithm

In numerical linear algebra, the QR algorithm is an eigenvalue algorithm: that is, a procedure to calculate the eigenvalues and eigenvectors of a matrix. The QR transformation was developed in the late 1950s by John G.F. Francis and by Vera N. Kublanovskaya , working independently...

.

If A has linearly independent columns (say n columns), then the first n columns of Q form an orthonormal basis

Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

for the column space

Column space

In linear algebra, the column space of a matrix is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of m-dimensional Euclidean space...

of A. More specifically, the first k columns of Q form an orthonormal basis for the span

Linear span

In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...

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 is responsible for the triangular form of R.

Square matrix

Any real square matrix A may be decomposed as

where Q is an orthogonal matrix

Orthogonal matrix

In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....

(its columns are orthogonal unit vectors meaning Q^TQ = I) and R is an upper triangular matrix

Triangular matrix

In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where either all the entries below or all the entries above the main diagonal are zero...

(also called right triangular matrix). This generalizes to a complex square matrix A and a unitary matrix Q. If A is invertible, then the factorization is unique if we require that the diagonal elements of R are positive.

Rectangular matrix

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 R₁ is an n×n upper triangular matrix, Q₁ is m×n, Q₂ is m×(m−n), and Q₁ and Q₂ both have orthogonal columns.

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

Cholesky decomposition

In linear algebra, the Cholesky decomposition or Cholesky triangle is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose. It was discovered by André-Louis Cholesky for real matrices...

of A A (= A^TA 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

Gram–Schmidt process

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 Rn...

, Householder transformation

Householder transformation

In linear algebra, a Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. Householder transformations are widely used in numerical linear algebra, to perform QR decompositions and in the first step of the QR algorithm...

s, or Givens rotations. Each has a number of advantages and disadvantages.

Using the Gram–Schmidt process

Consider the Gram–Schmidt process

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 then rearrange the equations above so that the

s are on the left, using the fact that the

are unit vectors:

where

. This can be written in matrix form:

where:

Example

Consider the decomposition of

Recall that an orthogonal matrix

has the property

Then, we can calculate

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.

Using Householder reflections

A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

or hyperplane

Hyperplane

A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...

. 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 ||

|| = |α| 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 As final upper triangular form, to avoid loss of significance

Loss of significance

Loss of significance is an undesirable effect in calculations using floating-point arithmetic. It occurs when an operation on two numbers increases relative error substantially more than it increases absolute error, for example in subtracting two large and nearly equal numbers. The effect is that...

. In the complex case, set

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

, where

where

is the conjugate transpose

Conjugate transpose

In mathematics, the conjugate transpose, Hermitian transpose, Hermitian conjugate, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry...

(transjugate) of

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 Q₁ we obtain when we choose the first matrix column for x. This results in a matrix Q₁A with zeros in the left column (except for the first row).

This can be repeated for A′ (obtained from Q₁A by deleting the first row and first column), resulting in a Householder matrix Q′₂. Note that Q′₂ is smaller than Q₁. Since we want it really to operate on Q₁A 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 a upper triangular matrix. So, with

is a QR decomposition of

.

This method has greater numerical stability

Numerical stability

In the mathematical subfield of numerical analysis, numerical stability is a desirable property of numerical algorithms. The precise definition of stability depends on the context, but it is related to the accuracy of the algorithm....

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

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

, to

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

Minor (linear algebra)

In linear algebra, a minor of a matrix A is the determinant of some smaller square matrix, cut down from A by removing one or more of its rows or columns...

, 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

The matrix Q is orthogonal and R is upper triangular, so A = QR is the required QR-decomposition.

Using Givens rotations

QR decompositions can also be computed with a series of Givens rotations. Each rotation zeros 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

Householder transformation

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

. 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 concatenation of all the Givens matrices

. Thus, we have

, and the QR decomposition is

Connection to a determinant or a product of eigenvalues

We can use QR decomposition to find the absolute value of the determinant

Determinant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

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 an unitary matrix.

From the properties of SVD

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

and determinant of matrix, we have

where

are singular values of

.

Note that the singular values of

and

are identical, although the complex eigenvalues of them may be different.
However, if A is square, it holds that

In conclusion, QR decomposition can be used efficiently to calculate a product of eigenvalues or singular values of matrix.

Column pivoting

QR decomposition with column pivoting introduces a permutation matrix

Permutation matrix

In mathematics, in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry 1 in each row and each column and 0s elsewhere...

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

Singular value decomposition

In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....

, forming the basis of so-called rank-revealing QR algorithms.

External links

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.
Into contains an open source implementation of QR decomposition in C++.

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Square matrix

Rectangular matrix

QL, RQ and LQ decompositions

Computing the QR decomposition

Using the Gram–Schmidt process

Example

Relation to RQ decomposition

Using Householder reflections

Example

Using Givens rotations

Example

Connection to a determinant or a product of eigenvalues

Column pivoting

See also

External links