Question

Let \(Z\) be an \(m \times n\) matrix such that \(Z^{H} Z=I_{n}\) (for example, \(Z\) is a unit column in \(\mathbb{C}^{n}\) ). a. Show that \(V=Z Z^{H}\) is hermitian and satisfies \(V^{2}=V\) b. Show that \(U=I-2 Z Z^{H}\) is both unitary and hermitian \(\left(\right.\) so \(\left.U^{-1}=U^{H}=U\right)\).

Short answer

Both conditions are satisfied: \(V\) is hermitian and idempotent; \(U\) is unitary and hermitian.

Step-by-step solution

Understanding the Hermitian Condition

Given matrix \(V = Z Z^H\), we need to show that \(V\) is Hermitian. By definition, a matrix is Hermitian if \(V = V^H\). Start by calculating \(V^H\). Since \(V = Z Z^H\), we have \(V^H = (Z Z^H)^H = (Z^H)^H Z^H = Z Z^H = V\). Thus, \(V\) is Hermitian.

Proving the Idempotent Property

Next, show that \(V^2 = V\). Compute \(V^2 = (Z Z^H)(Z Z^H) = Z (Z^H Z) Z^H\). Given \(Z^H Z = I_n\), it follows that \(V^2 = Z I_n Z^H = Z Z^H = V\). Therefore, \(V\) is idempotent, i.e., \(V^2 = V\).

Unitary Matrix Condition

For matrix \(U = I - 2Z Z^H\), we need to show that it is unitary, meaning \(U U^H = I\). Compute \(U^H = (I - 2Z Z^H)^H = I - 2(Z Z^H)^H = I - 2Z Z^H = U\). Thus, \(U\) is Hermitian and unitary if \(UU^H = (I - 2Z Z^H)(I - 2Z Z^H) = I\). Calculate \(UU = I - 4Z Z^H + 4Z (Z^H Z) Z^H = I - 4Z Z^H + 4Z I Z^H = I\). So, \(U\) is unitary.

Demonstrating Hermitian and Unitary equivalency

Since \(U^H = U\), \(U\) is Hermitian, as shown before. Since \(UU = I\), \(U\) is clearly unitary. This demonstrates \(U^{-1} = U^H = U\).

Key concepts

Idempotent Matrix

An idempotent matrix is a fascinating concept in linear algebra. Such a matrix, when multiplied by itself, yields the same matrix again. This peculiar property can be expressed by the equation \( A^2 = A \). In our exercise, we encountered this with the matrix \( V = Z Z^H \). We showed that \( V^2 = V \), implying that \( V \) is idempotent.

Here’s a simple way to visualize this: imagine an action performed once and repeating it has no further effect. For example, consider reflecting an object against a plane; doing it once returns a mirrored object, and doing it again brings it back to the original position. Similarly, idempotency means the operation (here, matrix multiplication) has a stabilizing effect.

For practical applications, idempotent matrices often arise in statistics, particularly in projection matrices used for linear regression. Whenever you see the term "project", think of idempotent matrices!

Unitary Matrix

A unitary matrix is a matrix that preserves the inner product. Essentially, this means it preserves lengths and angles, akin to a rigid transformation. The defining property of a unitary matrix \( U \) is that \( U U^H = U^H U = I \). In the given exercise, the matrix \( U = I - 2Z Z^H \) was both unitary and Hermitian.

Why does this matter? In mathematics and physics, unitary matrices often represent rotations and reflections. Think about the need to rotate a vector in space without altering its length; this is what a unitary transformation achieves.

One of the fantastic properties of unitary matrices is that their inverse is equal to their conjugate transpose, \( U^{-1} = U^H \). This makes them remarkably convenient for computations and theoretical proofs.

Applications? Look no further than quantum mechanics, where unitary matrices help describe the state's evolution over time while preserving probabilities.

Complex Matrices

Complex matrices consist of complex numbers, numbers of the form \( a + bi \), where \( i \) is the square root of \( -1 \). These matrices are crucial in advanced fields due to the versatility of complex numbers, granting more comprehensive solutions and insights.

Let's consider the matrix \( Z \) from our exercise, which resides in the complex number field \( \mathbb{C}^n \). The presence of the conjugate transpose, denoted by \( Z^H \), hints at its role in effectively handling complex elements. Conjugate transposition involves taking the transpose of the matrix and then replacing each element with its complex conjugate.

One of the key benefits of complex matrices is in simplifying and solving problems associated with wave functions in physics, like in electromagnetic theory or quantum mechanics. They're also crucial in engineering disciplines, such as systems design and signal processing.

By using complex matrices, we can embark on solving problems that real matrices wouldn't handle efficiently, making them indispensable in both theoretical and applied mathematics.