Question

a. Show that every complex matrix \(Z\) can be written uniquely in the form \(Z=A+i B,\) where \(A\) and \(B\) are real matrices. b. If \(Z=A+i B\) as in (a), show that \(Z\) is hermitian if and only if \(A\) is symmetric, and \(B\) is skewsymmetric (that is, \(B^{T}=-B\) ).

Short answer

Yes, every complex matrix can be uniquely decomposed, and it's hermitian if \( A \) is symmetric, and \( B \) is skew-symmetric.

Step-by-step solution

Understanding the Complex Matrix Decomposition

Let's start by considering a complex matrix \( Z \), which has elements of the form \( z_{ij} = a_{ij} + i b_{ij} \), where \( a_{ij} \) and \( b_{ij} \) are real numbers. We can decompose \( Z \) into two parts: a real part \( A \) containing all \( a_{ij} \) elements, and an imaginary part \( B \) containing all \( b_{ij} \) elements. This gives us the decomposition \( Z = A + iB \), where \( A \) and \( B \) are real matrices consisting of \( a_{ij} \) and \( b_{ij} \), respectively.

Ensuring Uniqueness of the Decomposition

To show that this decomposition is unique, assume that \( Z = A_1 + iB_1 = A_2 + iB_2 \). This implies \( A_1 + iB_1 = A_2 + iB_2 \), which can be separated into real and imaginary parts, giving us \( A_1 = A_2 \) and \( B_1 = B_2 \). This confirms that the real and imaginary parts of the matrix are unique.

Defining Hermitian Matrices

A matrix \( Z \) is hermitian if it satisfies the condition \( Z^* = Z \), where \( Z^* \) is the conjugate transpose of \( Z \). When \( Z = A + iB \), the hermitian condition becomes \( (A + iB)^* = A + iB \). The conjugate transpose transforms \( A + iB \) into \( A^T - iB^T \). So, for \( Z \) to be hermitian, \( A^T - iB^T = A + iB \) must hold.

Analyzing Conditions for Hermitian Matrix

Equating the real and imaginary parts from \( A^T - iB^T = A + iB \), we get two separate conditions: \( A^T = A \) and \( -B^T = B \). The first condition \( A^T = A \) implies that \( A \) is symmetric, while the second condition \( -B^T = B \) implies that \( B \) is skew-symmetric. Thus, for \( Z \) to be hermitian, \( A \) must be symmetric and \( B \) must be skew-symmetric.

Conclusion

We have shown that any complex matrix \( Z \) can uniquely be expressed as \( A + iB \), where \( A \) and \( B \) are real matrices. Furthermore, \( Z \) is hermitian if and only if \( A \) is symmetric and \( B \) is skew-symmetric.

Key concepts

Hermitian Matrices

Hermitian matrices are a special type of complex square matrix that hold a unique property: they are equal to their own conjugate transpose. Essentially, for any hermitian matrix \( Z \), the condition \( Z^* = Z \) must hold.

This means every element in the matrix has a specific relation to its corresponding element across the main diagonal. The conjugate transpose (or adjoint) \( Z^* \) involves taking the transpose of the matrix and replacing each element with its complex conjugate. This translates to the equation \( (A+iB)^* = A^T - iB^T \).

For hermitian matrices:

• The real part \( A \) must be symmetric, hence \( A = A^T \).
• The imaginary part \( B \) must be skew-symmetric, meaning \( B = -B^T \).
  
  

This unique combination of symmetry and skew-symmetry gives hermitian matrices their special properties and is foundational in quantum mechanics and various fields of engineering.

Symmetric Matrices

Symmetric matrices are a fundamental class of matrices characterized by their equal values across the main diagonal in both directions. In simpler terms, for a matrix to be symmetric, \( A = A^T \), meaning element \( a_{ij} = a_{ji} \) for all \( i \) and \( j \).

Here are some key characteristics:

• All entries that are mirrored across the diagonal are the same.
• Symmetric matrices are always real if they are part of a hermitian matrix's real part.
• They have real eigenvalues, and their eigenvectors can be orthogonally diagonalized.
  
  

Symmetric matrices play an important role in simplifying many mathematical frameworks, especially in numerical methods and optimization. They are directly tied to hermitian matrices, as the real part \( A \) of a hermitian matrix \( Z = A+iB \) must fulfill this symmetric condition.

Skew-Symmetric Matrices

Skew-symmetric matrices are intriguing mathematical objects, where each main diagonal entry is zero and the entries on either side of the main diagonal are negatives of each other. This means for a skew-symmetric matrix \( B \), we have \( B^T = -B \).

Key insights about skew-symmetric matrices include:

• All diagonal elements in a skew-symmetric matrix are zero.
• The off-diagonal elements satisfy \( b_{ij} = -b_{ji} \).
• The eigenvalues of skew-symmetric matrices are purely imaginary or zero.

In the case of a hermitian matrix, the imaginary part \( B \) must be skew-symmetric. This mirrors the critical balance within hermitian matrices, where the symmetric real part \( A \) pairs with the skew-symmetric imaginary part \( B \) to meet the hermitian condition. These matrices often emerge in physics and engineering, particularly in the study of rotational dynamics.