How Householder reflectors can be used to implement a QR decomposition?
The Householder reflection method of QR decomposition works by finding appropriate H matrices and multiplying them from the left by the original matrix A to construct the upper triangular matrix R.
How do you find q in household?
As we saw earlier, unlike the Gram-Schmidt procedure, the Householder reflection approach does not explicitly form the Q matrix. However, the Q matrix can be found by taking the dot product of each successively formed Householder matrix.
What is the use of QR decomposition?
QR decomposition is often used to solve the linear least squares problem and is the basis for a particular eigenvalue algorithm, the QR algorithm.
What does a Householder matrix do?
The Householder matrix (or elementary reflector) is a unitary matrix that is often used to transform another matrix into a simpler one. In particular, Householder matrices are often used to annihilate the entries below the main diagonal of a matrix.
What does the Householder transformation do?
Householder transformations are widely used in numerical linear algebra, for example, to annihilate the entries below the main diagonal of a matrix, to perform QR decompositions and in the first step of the QR algorithm. They are also widely used for transforming to a Hessenberg form.
What does the householder transformation do?
What is the determinant of a householder Matrix?
The determinant of a Householder reflector is , since the determinant of a matrix is the product of its eigenvalues, in this case one of which is with the remainder being (as in the previous point).
What does a Householder mean?
Definition of householder : a person who occupies a house or tenement alone or as the head of a household.
Is the identity a Householder matrix?
cannot be real, as the nonreal eigenvalues of a real matrix must appear in complex conjugate pairs. identity matrix, which in particular could be a Householder matrix!