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Calculate Householder Matrix for 0 Below

Reviewed by Calculator Editorial Team

This calculator helps you construct a Householder reflection matrix that zeros elements below a specified diagonal position. Householder matrices are fundamental in numerical linear algebra for QR decomposition and solving linear systems.

What is a Householder Matrix?

A Householder matrix is a square matrix that reflects vectors over a hyperplane. It's defined by a vector v and has the form:

H = I - 2vvᵀ / (vᵀv)

Where:

  • I is the identity matrix
  • v is the reflection vector
  • vᵀ is the transpose of v

Householder matrices are orthogonal (Hᵀ = H⁻¹) and symmetric (Hᵀ = H), making them useful for numerical computations where stability is important.

How to Calculate

To calculate a Householder matrix that zeros elements below a specified diagonal position:

  1. Identify the vector to reflect (typically a column of the matrix being transformed)
  2. Compute the Householder vector v
  3. Construct the matrix H using the formula above
  4. Apply H to the matrix to zero the specified elements

The calculator automates this process for you.

Formula

For a vector x = [x₁, x₂, ..., xₙ]ᵀ, the Householder vector is:

v = x + sign(x₁) ||x|| e₁

Where e₁ is the first standard basis vector and ||x|| is the Euclidean norm of x.

The resulting Householder matrix will reflect vectors over the hyperplane perpendicular to v, effectively zeroing elements below the specified diagonal position.

Worked Example

Let's calculate a Householder matrix for the vector [3, 4, 0, 0]ᵀ to zero elements below the first diagonal position:

  1. Compute the norm: ||x|| = √(3² + 4²) = 5
  2. Construct v: v = [3, 4, 0, 0] + sign(3)*5*[1, 0, 0, 0] = [8, 4, 0, 0]
  3. Compute vᵀv = 8² + 4² = 80
  4. Construct H: H = I - 2vvᵀ/80

The resulting matrix will have the form:

H = [ [0.05, 0.10, 0, 0], [0.10, 0.70, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] ]

Applications

Householder matrices are used in:

  • QR decomposition for solving linear systems
  • Eigenvalue algorithms
  • Least squares problems
  • Numerical optimization

They provide a stable way to perform orthogonal transformations in numerical computations.

FAQ

What is the difference between Householder and Givens rotations?
Householder transformations are more general and can zero multiple elements at once, while Givens rotations only affect two elements. Householder matrices are typically more efficient for larger problems.
When would I use a Householder matrix instead of a Givens rotation?
Use Householder matrices when you need to zero multiple elements simultaneously or when working with dense matrices. Givens rotations are better for sparse matrices or when you only need to zero two elements.
Is the Householder matrix always orthogonal?
Yes, by construction, Householder matrices are orthogonal (Hᵀ = H⁻¹) and symmetric (Hᵀ = H), which is important for numerical stability.