Question

A complex matrix \(B\) is called skewhermitian if \(B^{H}=-B\). a. Show that \(Z-Z^{H}\) is skew-hermitian for any square complex matrix \(Z\). b. If \(B\) is skew-hermitian, show that \(B^{2}\) and \(i B\) are hermitian. c. If \(B\) is skew-hermitian, show that the eigenvalues of \(B\) are pure imaginary ( \(i \lambda\) for real \(\lambda\) ). d. Show that every \(n \times n\) complex matrix \(Z\) can be written uniquely as \(Z=A+B,\) where \(A\) is hermitian and \(B\) is skew-hermitian.

Short answer

(a) Yes, skew-hermitian. (b) Yes, both hermitian. (c) Eigenvalues are purely imaginary. (d) Unique decomposition as Z = A + B.

Step-by-step solution

Prove Z-Z^H is Skew-Hermitian

To show that \(Z-Z^{H}\) is skew-hermitian, we need to demonstrate that \((Z-Z^{H})^{H} = -(Z-Z^{H})\). We compute the conjugate transpose:\[(Z-Z^{H})^{H} = Z^{H} - (Z^{H})^{H} = Z^{H} - Z.\]Since \(- (Z-Z^{H}) = Z^{H} - Z\), we conclude that \(Z-Z^{H}\) is skew-hermitian.

Show B^2 is Hermitian

Assuming \(B\) is skew-hermitian, we have \(B^{H} = -B\). We want to show \((B^{2})^{H} = B^{2}\). Compute:\[(B^{2})^{H} = (B \cdot B)^{H} = B^{H} \cdot B^{H} = (-B) \cdot (-B) = B^{2}.\]Thus, \(B^{2}\) is hermitian.

Show iB is Hermitian

Given \(B^{H} = -B\), we want to prove \((iB)^{H} = iB\). Compute:\[(iB)^{H} = (-i) B^{H} = (-i)(-B) = iB.\]Thus, \(iB\) is hermitian.

Eigenvalues of Skew-Hermitian Matrices

Let \(\lambda\) be an eigenvalue of \(B\) with eigenvector \(\mathbf{v}\), such that \(B\mathbf{v} = \lambda \mathbf{v}\). Then, \(\mathbf{v}^{H}B\mathbf{v} = \lambda\|\mathbf{v}\|^{2}\), and\[\mathbf{v}^{H}B\mathbf{v} = -\mathbf{v}^{H}B^{H}\mathbf{v} = -(B\mathbf{v})^{H}\mathbf{v} = -\overline{\lambda} \|\mathbf{v}\|^{2}.\]Thus, \(\lambda = -\overline{\lambda}\) implies \(\lambda\) is purely imaginary.

Decompose Z into Hermitian and Skew-Hermitian Parts

Let \(Z\) be any square complex matrix. Define \(A = \frac{Z + Z^{H}}{2}\) and \(B = \frac{Z - Z^{H}}{2}\). Show:- \(A^{H} = \left(\frac{Z + Z^{H}}{2}\right)^{H} = \frac{Z^{H} + Z}{2} = A.\) So, \(A\) is hermitian.- \(B^{H} = \left(\frac{Z - Z^{H}}{2}\right)^{H} = \frac{Z^{H} - Z}{2} = -B.\) So, \(B\) is skew-hermitian.Therefore, \(Z = A + B\) is a unique decomposition.

Key concepts

Hermitian Matrices

Hermitian matrices are a special class of matrices in linear algebra with distinctive properties. A Hermitian matrix, denoted by matrix \(A\), satisfies the condition that \(A^H = A\), where \(A^H\) represents the conjugate transpose of \(A\). This definition implies that the matrix is equal to its own conjugate transpose.

Some important characteristics of Hermitian matrices include:

• The diagonal elements of a Hermitian matrix are always real numbers because the conjugate of a real number is the number itself.
• The eigenvalues of a Hermitian matrix are always real. This is crucial in many fields such as quantum mechanics, where observable quantities are represented using Hermitian operators.
• Hermitian matrices are often associated with symmetry, as they relate to real-valued functions and operators.

When dealing with a complex matrix like in the exercise, the concepts of Hermitian matrices become relevant in understanding parts of the matrix's decomposition properties.

Eigenvalues

Eigenvalues are key in understanding matrix transformations. They provide insight into the properties of a matrix. For a matrix \(B\), an eigenvalue \(\lambda\) is a scalar such that \( B\mathbf{v} = \lambda\mathbf{v} \) for some nonzero vector \(\mathbf{v}\), called an eigenvector.

In the context of skew-Hermitian matrices, which satisfy \(B^H = -B\), the eigenvalues have a special property of being purely imaginary. Let's break down why this is significant:

• A purely imaginary number is one of the form \(i\lambda\), where \(\lambda\) is a real number.
• This property arises from the equation \(\lambda = -\overline{\lambda}\), implying \(\lambda + \overline{\lambda} = 0\), therefore making \(\lambda\) purely imaginary.
• The significance of imaginary eigenvalues in physics includes rotational transformations and stability analyses, as they often indicate oscillatory systems.

Eigenvalues are pivotal in understanding the nature and behavior of matrices, especially in more complex systems where Hermitian and skew-Hermitian properties define different matrix characteristics.

Complex Matrices

Complex matrices are matrices composed of complex numbers, making them a versatile tool in advanced mathematics and engineering. A complex number has two parts: a real part and an imaginary part, expressed as \(a + bi\), where \(i\) is the imaginary unit and \(a\) and \(b\) are real numbers.

These matrices enter various domains such as quantum mechanics, control theory, and signal processing due to their ability to effectively model and solve equations involving complex numbers. Key points include:

• Operations on complex matrices operating on the real and imaginary components independently, such as addition, multiplication, and scalar multiplication.
• The concept of a conjugate transpose, vital in defining Hermitian and skew-Hermitian matrices. It involves taking the complex conjugate of each element and switching rows with columns.
• Complex matrices can be challenging because they incorporate properties and operations not limited to traditional real matrices, involving complex number arithmetic.

Understanding these matrices' structural properties is essential in applications requiring complex number computations.

Matrix Decomposition

Matrix decomposition refers to the methods of breaking down a matrix into constituent parts, often to simplify operations like solving systems of equations, finding inverses, or analyzing matrix properties. In the exercise, we explore decomposing a complex matrix \(Z\) into Hermitian and skew-Hermitian parts, \(Z = A + B\).

Here's how this decomposition is structured:

• Any square complex matrix \(Z\) can be uniquely expressed as the sum of a Hermitian matrix \(A\) and a skew-Hermitian matrix \(B\).
• \(A\) can be found as \(A = \frac{Z + Z^H}{2}\), ensuring \(A^H = A\), which confirms it is Hermitian.
• \(B\) is obtained as \(B = \frac{Z - Z^H}{2}\), satisfying \(B^H = -B\), thus making \(B\) skew-Hermitian.

This decomposition is significant in linear algebra as it simplifies the analysis of complex matrices, allowing easier manipulation and study of their properties, and has practical applications in numerical methods and theoretical physics.