How do you create a householder Matrix?

How do you create a householder Matrix?

Compute Householder reflection H v i that zeros-out bi+2,i, bi+3, , …, bn,i and form H v i B = H v i A T . Recalling that H v i = H v i , take the transpose of H v i A T , and we have A H v i , a matrix in which the elements ai,i+2, ai,i+3, …, ai,k are zero. Compute A H v i implicitly using Equation 17.12.

What algorithm does Matlab use for QR factorization?

The CORDIC QR algorithm is given in the following MATLAB function, where A is an M-by-N real matrix, and niter is the number of CORDIC iterations. Output Q is an M-by-M orthogonal matrix, and R is an M-by-N upper-triangular matrix such that Q*R = A .

Is Householder matrix symmetric?

H=I−(uuT/β), they are not equal.

What is household matrix?

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. Definition.

What are the eigenvalues of a householder Matrix?

The Householder matrix Ha is symmetric, orthogonal, diagonalizable, and all its eigenvalues are 1’s except one which is -1. Moreover, it is idempotent: H2a=I. When Ha is applied to a vector x, it reflects x through hyperplane {z:aTz=0}.

How does MATLAB compute QR?

[ C , R ] = qr( S , B ) computes C = Q’*B and the upper-triangular factor R ….The size of the outputs depends on the size of m -by- n matrix A :

1. If m > n , then qr computes only the first n columns of Q and the first n rows of R .
2. If m <= n , then the economy-size decomposition is the same as the regular decomposition.

Is a householder matrix orthogonal?

Householder transformations are orthogonal transfor- mations (reflections) that can be used to similar effect. Reflection across the plane orthogonal to a unit normal vector v can be expressed in matrix form as H = I − 2vvT .

Is tridiagonal matrix a square matrix?

In linear algebra, a tridiagonal matrix is a band matrix that has nonzero elements on the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal only. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be done with the Lanczos algorithm.