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 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.

Cases and definitionsEdit

Square matrixEdit

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). 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  ).

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 ≤ kn.[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]

Rectangular matrixEdit

More generally, we can factor a complex m×n matrix A, with mn, 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 (mnn zero matrix, Q1 is m×n, Q2 is m×(mn), 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).

QL, RQ and LQ decompositionsEdit

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

Computing the QR decompositionEdit

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 processEdit

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

 

ExampleEdit

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 decompositionEdit

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 disadvantagesEdit

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, which is almost never the case.

Using Householder reflectionsEdit

 
Householder reflection for QR-decomposition: The goal is to find a linear transformation that changes the vector   into a vector of the 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 mn.

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[2]

 

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×m identity matrix, set

 

Or, if   is complex

 

  is an m-by-m Householder matrix, which is both symmetric and orthogonal (Hermitian and unitary in the complex case), 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.

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

 

ExampleEdit

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 disadvantagesEdit

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 rotationsEdit

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.

ExampleEdit

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 disadvantagesEdit

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 parallelizable than the Householder reflection technique.

Connection to a determinant or a product of eigenvaluesEdit

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   is unitary,  . Thus,

 

where the   are the entries on the diagonal of  . Furthermore, because the determinant equals the product of the eigenvalues, we have

 

where the   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   denotes the zero matrix and   is a unitary matrix.

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

 

where the   are the 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 pivotingEdit

Pivoted QR differs from ordinary Gram-Schmidt in that it takes the largest remaining column at the beginning of each new step—column pivoting—[3] 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 problemsEdit

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   has dimensions   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:  . 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  :  . 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.)

GeneralizationsEdit

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

See alsoEdit

ReferencesEdit

  1. ^ a b c Trefethen, Lloyd N.; Bau, David III (1997). Numerical linear algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 978-0-898713-61-9.
  2. ^ Stoer, Josef; Bulirsch, Roland (2002), Introduction to Numerical Analysis (3rd ed.), Springer, p. 225, ISBN 0-387-95452-X
  3. ^ Strang, Gilbert (2019). Linear Algebra and Learning from Data (1st ed.). Wellesley: Wellesley Cambridge Press. p. 143. ISBN 978-0-692-19638-0.

Further readingEdit

External linksEdit