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So to sum it up this, the Householder QR algorithm has really excellent stability. It works for most matrices that you can imagine, as long as they have full column rank. For square matrices, the computational complexity of this householder QR is only a few times worse than LU decomposition. Householder transformation and QR decomposition A Householder transformation of a vector is its reflection with respect a plane (or hyperplane) through the origin represented by its normal vector of unit length, which can be found as where is the projection of onto. In general, the projection of onto any vector is. 4 QR decomposition and Householder transformations¶ Course: Math 535 - Mathematical Methods in Data Science (MMiDS) Author: Sebastien Roch, Department of Mathematics, University of Wisconsin-Madison. Recall the strategy used in deriving method for QR decomposition using Householder reflections. C) For the matrix A = 0.00001. eye (n) + hilb (n), where eye (n) returns an identity matrix and hilb (n) return a hilbert matrix of size n. 1) Apply QR decomposition for the above matrix using Gram-Schmidt, Householder reflections and method in b). These matrices are the so-called Householder reflections. Value List with two matrices Q and R, Q orthonormal and R upper triangular, such that A=Q%.%R.
Jacobi iteration Up:Solving Linear Algebraic Equations Previous:SVD for solving linearHouseholder transformation and QR decomposition
A Householder transformation of a vector is its reflection withrespect a plane (or hyperplane) through the origin represented by its normal vector of unit length , which can be found as
where is the projection of onto .In general, the projection of onto any vector is
The reflection of can be considered as a linear transformation represented by a matrix applied to :
where is a Householder matrix.
Obviously is symmetric
and it is also the inverse of its own:
We therefore have
i.e., is an orthogonal matrix (satisfying ).
As any vector can be normalized tohave a unit norm, the Householder matrix defined above can be written as:
We define specifically a vector
where is any N-D vector and is the first standard basis vector with all elements equals zero except the firstone. The norm squared of this vector can be found to be
When the Householder matrix based on this particular vector is applied to the vector itself, it produces its reflection
We see that all elements in except the first one are eliminated to zero.This feature of the Householder transformation is the reason why it is widely used.
The Householder transformation finds many applications in numerical computation.For example, it can be used to convert a given matrix into either a bidiagonal or tridiagonal form, which is needed in the algorithms for solving SVD and eigenvalue problems. The Householder transformation can also be used to carry out QR decompositionof an by square matrix :
where is an orthogonal matrix and is an uppertriangular matrix. Specifically, we first construct a Householder matrix based on the first column vector of, i.e., , by which the last elements of the first column of will become zero:
We then construct second transformation matrix
where is an ()-dimensional Householder matrix constructed based on the first column of , to zero the last elements of this column:
This process is repeated with the kth transformation matrix
where is an ()-dimensional Householder matrix constructedbased on the first column of the sub-matrix of the same dimensionobtained from the previous step.
After such iterations becomes an upper triangular matrix:
Pre-multiplying on both sides we get QR decomposition of :
Here satisfies
i.e., is orthogonal and so is .
QR decomposition is widely used in different algorithms (e.g., SVD, eigenvalue problems, etc.), and it can also be used to solve the linear system :
where can be obtained as:
Then we can find by solving
As is an upper triangular matrix, can be obtained by back-substitution.
The Householder transformation can also be used to carry out tridiagonalizationof an by matrix . Specifically we first define an matrix as shown below:
where the lower-right part of is an dimensional Householder matrixbased on an dimensional vector containing the last elements of the first column of . We first pre-multiply to to get:
where is an dimensional matrix containing the lower-right by elements of , and the lower-left block of the resultingmatrix only has a non-zero element as the result of the application of the Householder matrix. We next post-multiply to the result above to get
We next multiply both sides of the above by
to get
This process is recursively repeated until after iterations the by matrix is converted into a tridiagonal matrix.
Qr Householder
Qr Householder Pokemon
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Up:Solving Linear Algebraic Equations Previous:SVD for solving linearRuye Wang2015-02-12