QR decomposition

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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 Branch of mathematics

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.

Matrix (mathematics) Two-dimensional array of numbers with specific operations

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:

Contents

Cases and definitions

Square matrix

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.

Unit vector vector of length one

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.

Triangular matrix special kind of square matrix

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 Rn 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 Rn 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.

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 (mn) 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, 0 is an (m  nn 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. [1] 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 (= ATA 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.....

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 (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

Example

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

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.

Using Householder reflections

Householder reflection for QR-decomposition: The goal is to find a linear transformation that changes the vector
x
{\displaystyle x}
into a vector of same length which is collinear to
e
1
{\displaystyle e_{1}}
. We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors
x
{\displaystyle x}
and
e
1
{\displaystyle e_{1}}
are close to orthogonal. Instead, the Householder reflection reflects through the dotted line (chosen to bisect the angle between
x
{\displaystyle x}
and
e
1
{\displaystyle e_{1}}
). The maximum angle with this transform is 45 degrees. Householder.svg
Householder reflection for QR-decomposition: The goal is to find a linear transformation that changes the vector into a vector of same length which is collinear to . We could use an orthogonal projection (Gram-Schmidt) but this will be numerically unstable if the vectors and are close to orthogonal. Instead, the Householder reflection reflects through the dotted line (chosen to bisect the angle between and ). The maximum angle with this transform is 45 degrees.

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 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 Q2. Note that Q2 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 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.

OperationNumber 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

Example

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.

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

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 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 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.

Connection to a determinant or a product of eigenvalues

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.

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- [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.

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 [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.)

Generalizations

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

See also

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References

  1. 1 2 3 L. N. Trefethen and D. Bau, Numerical Linear Algebra (SIAM, 1997).
  2. Strang, Gilbert (2019). Linear Algebra and Learning from Data (1 ed.). Wellesley: Wellesley Cambridge Press. p. 143. ISBN   978-0-692-19638-0.

Further reading