Question

Show that a real Hermitian matrix is symmetric. Show that a real unitary matrix is orthogonal.

Short answer

A real Hermitian matrix is symmetric because it equals its transpose. A real unitary matrix is orthogonal because its transpose is its inverse.

Step-by-step solution

Define Hermitian Matrix

A matrix is Hermitian if it is equal to its own conjugate transpose. For a real Hermitian matrix, the matrix elements are real numbers, so the conjugate transpose is simply the transpose.

Transpose of a Real Hermitian Matrix

Let the real Hermitian matrix be denoted as \(\textbf{A}\). Since it is Hermitian, it must satisfy \(\textbf{A} = \textbf{A}^\text{H}\) where \(\textbf{A}^\text{H}\) is the conjugate transpose. For real matrices, the conjugate transpose is just the transpose: \(\textbf{A}^\text{T}\). Thus, \(\textbf{A} = \textbf{A}^\text{T}\).

Conclusion for Symmetric Property

Since \(\textbf{A} = \textbf{A}^\text{T}\), the real Hermitian matrix is symmetric.

Define Unitary Matrix

A matrix is unitary if its conjugate transpose is also its inverse, i.e., \(\textbf{U}^\text{H} \textbf{U} = \textbf{U} \textbf{U}^\text{H} = \textbf{I}\), where \(\textbf{U}^\text{H}\) is the conjugate transpose of \(\textbf{U}\).

Condition for Real Unitary Matrix

For a real unitary matrix \(\textbf{U}\), since its elements are real numbers, \(\textbf{U}^\text{H}\) reduces to its transpose \(\textbf{U}^\text{T}\). Therefore, the condition becomes \(\textbf{U}^\text{T} \textbf{U} = \textbf{I}\).

Conclusion for Orthogonal Property

If \(\textbf{U}^\text{T} \textbf{U} = \textbf{I}\), the real unitary matrix \(\textbf{U}\) is orthogonal.

Key concepts

Hermitian Matrix

A Hermitian matrix is a square matrix that is equal to its own conjugate transpose. Conjugate transpose, also known as Hermitian transpose, involves taking the transpose of the matrix and then taking the complex conjugate of each element. This means that for any matrix \(\textbf{A}\), it is Hermitian if \(\textbf{A} = \textbf{A}^\text{H}\). In the case of real matrices, the complex conjugate operation does not change the matrix elements since they are already real numbers. Therefore, a real Hermitian matrix must simply satisfy \(\textbf{A} = \textbf{A}^\text{T}\). This property makes any real Hermitian matrix symmetric, as it is equal to its transpose.

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. For any matrix \(\textbf{A}\), it is considered symmetric if \(\textbf{A} = \textbf{A}^\text{T}\). This means that the elements of the matrix are symmetrical along the main diagonal:

• Elements above the diagonal mirror those below it.
• Diagonal elements remain unchanged when the matrix is transposed.

This property is a direct consequence of the definition and is particularly important in many areas of mathematics and physics. An important point to remember is that all real Hermitian matrices are symmetric, but not all symmetric matrices are Hermitian since they do not necessarily involve complex conjugation.

Unitary Matrix

A unitary matrix is a complex matrix whose conjugate transpose is also its inverse. For a matrix \(\textbf{U}\) to be unitary, it must satisfy the condition \(\textbf{U}^\text{H} \textbf{U} = \textbf{I}\), where \(\textbf{U}^\text{H}\) is the conjugate transpose and \(\textbf{I}\) is the identity matrix. This implies that multiplying the matrix by its conjugate transpose results in the identity matrix. In the special case of real matrices, the conjugate transpose simplifies to just the transpose. Thus, for a real unitary matrix, \(\textbf{U}^\text{H} = \textbf{U}^\text{T}\), and the condition becomes \(\textbf{U}^\text{T} \textbf{U} = \textbf{I}\).

Orthogonal Matrix

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthonormal vectors. For a matrix \(\textbf{Q}\) to be orthogonal, it must satisfy \(\textbf{Q}^\text{T} \textbf{Q} = \textbf{I}\), where \(\textbf{Q}^\text{T}\) is the transpose of \(\textbf{Q}\) and \(\textbf{I}\) is the identity matrix. This condition implies that the matrix's transpose is its inverse. Characteristics of orthogonal matrices include:

• Their determinant must be \(\text{±1}\).
• They preserve the dot product of vectors.
• Multiplication by an orthogonal matrix preserves vector lengths and angles between vectors.

Notably, a real unitary matrix is simply an orthogonal matrix.