Question

(a) For a given real vector \(\mathbf{u}\) satisfying \(\|\mathbf{u}\|_{2}=1\), show that the matrix \(P=I-2 \mathbf{u u}^{T}\) is orthogonal. (b) Suppose \(A\) is a complex-valued matrix. Construct a complex analogue of Householder transformations, with the reflector given by \(P=I-2 \mathrm{uu}^{*}\), where \(*\) denotes a complex conjugate transpose and \(\mathbf{u}^{*} \mathbf{u}=1\). (The matrix \(P\) is now unitary, meaning that \(P^{*} P=I .\) )

Short answer

Question: Prove that the matrix \(P=I-2 \mathbf{u u}^{T}\) is orthogonal, where \(\mathbf{u}\) is a real vector with \(\|\mathbf{u}\|_{2}=1\), and construct a complex analogue of Householder transformations with the reflector being unitary. Answer: To prove that \(P\) is orthogonal, we show that \(P^TP=PP^T=I\). After computing and simplifying the products, we find that \(P^TP=PP^T=I\), thus proving that \(P\) is orthogonal. For the complex analogue, we construct the reflector operator \(P=I-2 \mathbf{u u}^{*}\) such that \(P\) is unitary. We compute and simplify the products \(P^*P\) and \(PP^*\) to find that \(P^*P=PP^*=I\), which proves that \(P\) is unitary and forms a complex analogue of Householder transformations.

Step-by-step solution

Definition of orthogonal matrix and properties

An orthogonal matrix is defined as a real square matrix that has its columns and rows orthogonal, i.e., for matrix \(S\), \(S^TS=I\) and \(ST^T=I\). Our job is to show that this is the case with the matrix \(P=I-2 \mathbf{u u}^{T}\).

Compute \(P^T P\) and \(P P^T\)

In order to check whether \(P\) is orthogonal, we need to compute the product \(P^TP\) and \(PP^T\). So we first write down \(P^T\) as \((I-2 \mathbf{u u}^{T})^T= I - 2(\mathbf{u u}^{T})^T = I - 2\mathbf{u}^{T}\mathbf{u}\). Now we compute the products: \(P^{T}P=(I-2\mathbf{u}^{T}\mathbf{u})(I-2\mathbf{u}\mathbf{u}^{T})\) \(PP^{T}=(I-2\mathbf{u}\mathbf{u}^{T})(I-2\mathbf{u}^{T}\mathbf{u})\)

Simplify the products

Now, we will simplify the product expressions using the properties \(\mathbf{u}^{T}\mathbf{u}=1\) and \(\mathbf{u}\mathbf{u}^{T}\mathbf{u}=\mathbf{u}\), and \((\mathbf{u}\mathbf{u}^{T})(\mathbf{u}\mathbf{u}^{T})=\mathbf{u}\mathbf{u}^{T}\): \(P^{T}P=I-2\mathbf{u}\mathbf{u}^{T}-2\mathbf{u}^{T}\mathbf{u}+4\mathbf{u}\mathbf{u}^{T}\mathbf{u}\mathbf{u}^{T}=I-2\mathbf{u}\mathbf{u}^{T}-2\mathbf{u}\mathbf{u}^{T}+4\mathbf{u}\mathbf{u}^{T}=I\) \(PP^{T}=I-2\mathbf{u}\mathbf{u}^{T}-2\mathbf{u}\mathbf{u}^{T}+4\mathbf{u}\mathbf{u}^{T}\mathbf{u}\mathbf{u}^{T}=I-2\mathbf{u}\mathbf{u}^{T}-2\mathbf{u}\mathbf{u}^{T}+4\mathbf{u}\mathbf{u}^{T}=I\) Since \(P^TP=PP^T=I\), we have proven that the matrix \(P\) is orthogonal. For part (b):

Definition of unitary matrix and properties

A unitary matrix is a complex square matrix that has its columns and rows orthogonal in the complex sense, i.e., for matrix \(S\), \(S^*S=I\) and \(S S^*=I\). Our job is to construct the reflector operator \(P=I-2 \mathbf{u u}^{*}\) such that \(P\) is unitary.

Compute \(P^* P\) and \(P P^*\)

To check whether \(P\) is unitary, we need to compute the product \(P^*P\) and \(PP^*\). So we first write down \(P^*\) as \((I-2\mathbf{u u}^{*})^*=I-2(\mathbf{u u}^{*})^* = I - 2\underline{\mathbf{u}^{*} \mathbf{u}}\). Now, we compute the products: \(P^{*}P=(I-2\mathbf{u}^{*}\mathbf{u})(I-2\mathbf{u}\mathbf{u}^{*})\) \(PP^{*}=(I-2\mathbf{u}\mathbf{u}^{*})(I-2\mathbf{u}^{*}\mathbf{u})\)

Simplify the products

Now, we will simplify the product expressions using the property \(\mathbf{u}^{*}\mathbf{u}=1\), just like we did in part (a): \(P^{*}P=I-2\mathbf{u}\mathbf{u}^{*}-2\mathbf{u}^{*}\mathbf{u}+4\mathbf{u}\mathbf{u}^{*}\mathbf{u}^{*}\mathbf{u}=I-2\mathbf{u}\mathbf{u}^{*}-2\mathbf{u}\mathbf{u}^{*}+4\mathbf{u}\mathbf{u}^{*}=I\) \(PP^{*}=I-2\mathbf{u}\mathbf{u}^{*}-2\mathbf{u}^{*}\mathbf{u}+4\mathbf{u}\mathbf{u}^{*}\mathbf{u}^{*}\mathbf{u}=I-2\mathbf{u}\mathbf{u}^{*}-2\mathbf{u}\mathbf{u}^{*}+4\mathbf{u}\mathbf{u}^{*}=I\) Since \(P^*P=PP^*=I\), we have constructed the reflector \(P\), which is unitary, and forms a complex analogue of Householder transformations.

Key concepts

Orthogonal Matrix

An orthogonal matrix is a special type of square matrix. These matrices are famous for preserving the standard inner product, which means they keep vector lengths and angles the same before and after the transformation. One key aspect of orthogonal matrices is their defining property: the transpose of an orthogonal matrix is also its inverse. This can be expressed mathematically as: \(S^TS=I\), where \(S\) is the matrix and \(I\) is the identity matrix.

To understand this, let's think about what the transpose and inverse mean. The transpose \(S^T\) flips the matrix over its diagonal, while the inverse \(S^{-1}\) is like the matrix that "undoes" the transformation of \(S\). If a matrix is orthogonal, multiplying it by its transpose results in the identity matrix. This illustrates that all columns and rows of \(S\) are orthogonal (i.e., perpendicular) and normalized (i.e., they have a length of one), which is why the transformations they represent won't change a vector's length or angle.

In the exercise, the matrix \(P = I - 2 \mathbf{uu}^T\) was shown to be orthogonal by verifying that \(P^TP = PP^T = I\). This highlights that the matrix \(P\), generated in this way from a normalized vector \(\mathbf{u}\), satisfies the orthogonality conditions.

Unitary Matrix

A unitary matrix is essentially the complex brother of an orthogonal matrix. This type of matrix is particularly significant in contexts involving complex numbers, which are ubiquitous in fields like quantum mechanics. The fundamental characteristic of a unitary matrix is described by its ability to preserve inner products over the complex field, hence maintaining vector norms and angles.

The definition of a unitary matrix involves the complex conjugate transpose, often denoted by an asterisk (*). For any unitary matrix \(S\), the relation \(S^*S = I\) and \(SS^* = I\) holds. This ensures that the matrix retains the unitary property whether acting as an initial transformation or its reversal. Just like their real counterparts (orthogonal matrices), columns (and rows) of a unitary matrix are orthogonal and are normed to 1.

In part (b) of the exercise, the construction of a matrix \(P = I - 2 \mathbf{uu}^*\) was needed to be verified as unitary. By showing \(P^*P = PP^* = I\), it was successfully determined that this transformation forms a unitary matrix, preserving essential properties even in the complex space.

Complex Conjugate Transpose

The concept of a complex conjugate transpose is pivotal when dealing with matrices with complex numbers. The operation involves two transformations on a matrix: one being the transpose, where you flip the matrix over its diagonal, and the other being taking the complex conjugate of each entry.

Given a matrix \(A\), the complex conjugate of \(A\) is achieved by changing the sign of the imaginary part of all complex numbers in \(A\). Following this, the transpose operation is applied, resulting in the complex conjugate transpose, denoted as \(A^*\). Mathematically, this is expressed as \((A^*)_{ij} = \overline{A_{ji}}\), where \(\overline{A_{ji}}\) represents the complex conjugate of the entry in the \(i^{th}\) row and \(j^{th}\) column.

This transformation is instrumental when assessing unitary matrices, as demonstrated in the exercise for part (b). It was essential to work through the conjugate transpose to confirm the unitary status of matrix \(P = I - 2 \mathbf{uu}^*\), maintaining all necessary properties expected of such transformations.

Reflector Operator

A reflector operator is an efficient mathematical tool that enables the reflection of vectors over hyperplanes. The Householder transformation, which the exercise is based on, is a common example of a reflector in linear algebra.

The formula for the Householder reflector can be seen as \(P = I - 2 \mathbf{uu}^T\) in the real-valued case, or \(P = I - 2 \mathbf{uu}^*\) for complex matrices. Here, \(\mathbf{u}\) is a vector which essentially represents the hyperplane over which the reflection occurs, and it is normalized to have a length (norm) of 1.

This kind of transformation is central to various algorithms in numerical linear algebra, particularly those needed for orthogonalization, QR decomposition, and solving linear systems. The concept of reflection essentially allows a vector's negative component with respect to the hyperplane, defined by the vectors, to be "flipped" in directionality. This can be very efficient in transforming a given matrix into an easier-to-handle form for computation.

The exercise's context neatly illustrates how the reflector operator can be adapted from real numbers to complex numbers, allowing for the application of reflections in a broader scope of mathematical problems.