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In linear algebra, the QR decomposition (also called the QR factorization) of a matrix is a decomposition of the matrix into an orthogonal and a triangular matrix. The QR decomposition is often used to solve the linear least squares problem. The QR decomposition is also the basis for a particular eigenvalue algorithm, the QR algorithm. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...
In the mathematical discipline of linear algebra, a matrix decomposition is a factorization of a matrix into some canonical form. ...
In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
Linear least squares is a mathematical optimization technique to find an approximate solution for a system of linear equations that has no exact solution. ...
It has been suggested that this article or section be merged with Symbolic computation of matrix eigenvalues. ...
A QR algorithm is a procedure to calculate the eigenvalues of a matrix. ...
Definition
A QR decomposition of a real square matrix A is a decomposition of A as  where Q is an orthogonal matrix (meaning that QTQ = I ) and R is an upper triangular matrix. Analogously, we can define the QL, RQ, and LQ decompositions of A. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...
In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
More generally, we can factor a complex m×n matrix (with m ≥ n) as the product of an m×n unitary matrix (in the sense that Q∗Q = I ) and an n×n upper triangular matrix. In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse...
If A is nonsingular, then this factorization is unique if we require that the diagonal elements of R are positive. In linear algebra, an n-by-n (square) matrix is called invertible or non-singular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ...
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. In mathematics and numerical analysis, the Gram–Schmidt Process is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
In mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane. ...
In mathematics, a Givens rotation is a matrix of the form where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. ...
Computing QR by means of Gram-Schmidt Consider the Gram–Schmidt process, with the vectors to be considered in the process as columns of the matrix  . We define  In mathematics and numerical analysis, the Gram–Schmidt Process is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
Then
     Naturally then, we rearrange the equations so the ais are the subject, to get the following     Each of these projections of the vectors  onto one of these ej are merely the inner product of the two, since the vectors are normed. In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ...
Now these equations can be written in matrix form, viz.,
 But the product of each row and column of the matrices above give us a respective column of A that we started with, and together, they give us the matrix A, so we have factorized A into an orthogonal matrix Q (the matrix of eks), via Gram Schmidt, and the obvious upper triangular matrix as a remainder R. In the mathematical discipline of linear algebra, a triangular matrix is a special kind of square matrix where the entries below or above the main diagonal are zero. ...
Alternatively,  can be calculated as follows:
Recall that  Then, we have
 Note that   and QQT = I, so QT = Q − 1.
Example Consider the decomposition of  Recall the orthogonal matrix Q such that  Then, we can calculate Q by means of Gram-Schmidt as follows:   Thus, we have   Performing this operation using MATLAB, including numerical errors due to finite precision operation, leads to:  MATLAB is a numerical computing environment and programming language. ...
Computing QR by means of Householder reflections A Householder reflection (or Householder transformation) is a transformation that takes a vector and reflects it about some plane. We can use this property to calculate the QR factorization of a matrix. In mathematics, a Householder transformation in 3-dimensional space is the reflection of a vector in a plane. ...
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 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 first coordinate of to avoid loss of significance. If is a complex vector, then the definition A floating-point number is a digital representation for a number in a certain subset of the rational numbers, and is often used to approximate an arbitrary real number on a computer. ...
Loss of significance is an undesirable effect in calculations using floating-point arithmetic. ...
 should be used (Stoer & Bulirsch 2002, p. 225).
Then, where  is the vector (1,0,...,0)T, and ||·|| the Euclidean norm, set Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
   Q is a 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 QA with zeros in the left column (except for the first row). For alternate meanings, such as the musical instrument, see triangle (disambiguation). ...
 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 t iterations of this process, t = min(m − 1,n),  is a upper triangular matrix. So, with  A = QR is a QR decomposition of A.
This method has greater numerical stability than the Gram-Schmidt method above. In the mathematical subfield of numerical analysis, numerical stability is a property of numerical algorithms. ...
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 | 2(n − k + 1)2 | | additions | (n − k + 1)2 + (n − k + 1)(n − k) + 2 | | division | 1 | | square root | 1 | Summing these numbers over the (n − 1) steps (for a square matrix of size n), the complexity of the algorithm 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, - α = 14 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 In linear algebra, a minor of a matrix is the determinant of a certain smaller matrix. ...
 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.
Computing QR by means of 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 mathematics, a Givens rotation is a matrix of the form where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. ...
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 G1. We will first rotate the vector (6, − 4), to point along the X axis. This vector has an angle  . We create the orthogonal Givens rotation matrix, G1: In mathematics, a Givens rotation is a matrix of the form where c = cos(θ) and s = sin(θ) appear in the i-th / k-th row and column, respectively. ...
  And the result of G1A now has a zero in the  element.  We can similarly form Givens matrices G2 and G3, which will zero the sub-diagonal elements a21 and a32, forming a triangular matrix R. The orthogonal matrix QT is formed from the concatenation of all the Givens matrices QT = G3G2G1. Thus, we have G3G2G1A = QTA = R, and the QR decomposition is A = QR.
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 A = QR. Then we have In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ...
 Since Q is unitary, | det(Q) | = 1. Thus,  where rii are the entries on the diagonal of R.
Furthermore, because the determinant equals the product of the eigenvalues, we have
 where λi are eigenvalues of A.
We can extend the above properties to non-square complex matrix A 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 O is a zero matrix and Q is an unitary matrix.
From the properties of SVD and determinant of matrix, we have In linear algebra, the singular value decomposition (SVD) is an important factorization of a rectangular real or complex matrix, with several applications in signal processing and statistics. ...
 where σi are singular values of A.
Note that the singular values of A and R 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.
See also In mathematics, particularly in linear algebra and functional analysis, the polar decomposition of a matrix or linear operator is a factorization analogous to polar decomposition of a nonzero complex number z where r is the absolute value of z (a positive real number), and is the complex sign of z. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ...
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is a collection of results about linear operators or about matrices. ...
References - Horn, Roger A. & Charles R. Johnson (1985), Matrix Analysis, Cambridge University Press, ISBN 0-521-38632-2. Section 2.8.
- Stoer, Josef & Roland Bulirsch (2002), Introduction to Numerical Analysis (3rd ed.), Springer, ISBN 0-387-95452-X.
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