(a) Write down the Householder matrices corresponding to the...

(a) Write down the Householder matrices corresponding to the following unit vectors: (i) $(1,0)^T$, (ii) $\left(\frac{3}{5}, \frac{4}{5}\right)^T$, (iii) $(0,1,0)^T,(i v)\left(\frac{1}{\sqrt{2}}, 0,-\frac{1}{\sqrt{2}}\right)^T$. (b) Find all vectors fixed by a Householder matrix, i.e., $H \mathbf{v}=\mathbf{v}$ - first for the matrices in part (a), and then in general. (c) Is a Householder matrix a proper or improper orthogonal matrix?

(a) Write down the Householder matrices corresponding to the following unit vectors:
(i) $(1,0)^T$,
(ii) $\left(\frac{3}{5}, \frac{4}{5}\right)^T$,
(iii)
$(0,1,0)^T,(i v)\left(\frac{1}{\sqrt{2}}, 0,-\frac{1}{\sqrt{2}}\right)^T$.
(b) Find all vectors fixed by a Householder matrix, i.e., $H \mathbf{v}=\mathbf{v}$ - first for the matrices in part (a), and then in general. (c) Is a Householder matrix a proper or improper orthogonal matrix?

Key Concepts

Fixed Vectors and Invariant Subspaces

The fixed vectors of a Householder matrix are those vectors that remain unchanged by the reflection. Specifically, any vector that lies in the hyperplane orthogonal to the unit vector u (used in the construction of the Householder matrix) will be fixed, i.e., Hv = v. This subspace of fixed vectors plays an important role in understanding the effect of the transformation on the space.

Orthogonal Matrices and Proper/Improper Classification

Orthogonal matrices are matrices that satisfy H^T H = I, indicating they preserve the length of vectors upon transformation. They can be classified as proper orthogonal matrices if they have a determinant of +1, which means they represent rotations, or improper orthogonal matrices if the determinant is -1, which corresponds to reflections. Householder matrices, having determinant -1, are therefore considered improper orthogonal matrices.

Householder Matrix

A Householder matrix is a type of orthogonal matrix used primarily for reflecting vectors about a plane or hyperplane. It is typically constructed in the form H = I - 2uu^T, where u is a unit vector. The construction yields a symmetric matrix that is both orthogonal and involutory (meaning H^2 = I), making it an efficient tool for transforming vectors without changing their length.

Householder Reflection

Householder reflections are the geometric transformations represented by Householder matrices. They reflect a vector through a hyperplane that is orthogonal to a given unit vector u. This reflection property is utilized in many numerical algorithms, such as QR decomposition, for systematically zeroing components of vectors while preserving the Euclidean norm.

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