Question

(a) Show that the product of two symmetric matrices is symmetric if and only if they commute. (b) When is the product of two Hermitian matrices a Hermitian matrix?

Short answer

(a) The product of two symmetric matrices is symmetric if and only if they commute. (b) The product of two Hermitian matrices is Hermitian if and only if they commute.

Step-by-step solution

- Define Symmetric Matrices

A matrix A is symmetric if it equals its transpose, i.e., \( A = A^T \). Similarly, a matrix B is symmetric if \( B = B^T \).

- Product of Two Matrices

Consider the product \(C = AB\) of two symmetric matrices A and B. For C to be symmetric, \( C = C^T \) must hold true.

- Transpose of the Product

Take the transpose of the product matrix C: \( C^T = (AB)^T \). Using the property of transposes, this equals \( B^T A^T \).

- Substitute Symmetric Properties

Since A and B are symmetric, \( B^T = B \) and \( A^T = A \). Thus, \( C^T = BA \).

- Requirement for Symmetry

For C to be symmetric (\( C = AB \) and \( C^T = BA \)), \( AB = BA \) must hold true. Therefore, A and B must commute.

- Conclusion (Part a)

The product of two symmetric matrices is symmetric if and only if they commute. Hence, the statement is proven.

- Define Hermitian Matrices

A matrix A is Hermitian if it equals its conjugate transpose, i.e., \( A = A^* \). Similarly, a matrix B is Hermitian if \( B = B^* \).

- Product of Hermitian Matrices

Consider the product \( C = AB \) of two Hermitian matrices A and B. For C to be Hermitian, \( C = C^* \) must hold true.

- Conjugate Transpose of the Product

Take the conjugate transpose of the product matrix C: \( C^* = (AB)^* \). Using the property of conjugate transposes, this equals \( B^* A^* \).

- Substitute Hermitian Properties

Since A and B are Hermitian, \( A^* = A \) and \( B^* = B \). Thus, \( C^* = BA \).

- Requirement for Hermitian Product

For C to be Hermitian (\( C = AB \) and \( C^* = BA \)), \( AB = BA \) must hold true. Therefore, A and B must commute for the product to be Hermitian.

- Conclusion (Part b)

The product of two Hermitian matrices is Hermitian if and only if the matrices commute.

Key concepts

symmetric matrices

Symmetric matrices are special types of square matrices where the matrix is equal to its transpose. This means that for a symmetric matrix A, the condition is given by:
\[ A = A^T \].
This implies that the elements of the matrix are mirrored along the diagonal. In other words, the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column.
For example, consider the following symmetric matrix:
\[ \begin{pmatrix} 1 & 2 & 3 \ 2 & 4 & 5 \ 3 & 5 & 6 \right) \]
You can see that each element on the left side of the main diagonal (from top-left to bottom-right) is mirrored to the right side. This symmetry property ensures that the transpose of the matrix is the same as the original matrix.
Symmetric matrices are widely used in various fields such as physics, computer graphics, and statistics due to their unique properties.

matrix transpose

The transpose of a matrix is formed by swapping its rows with its columns. For a given matrix A, its transpose is denoted by \( A^T \).
This means that if the element in the i-th row and j-th column of matrix A is denoted by \( a_{ij} \), then the element in the i-th row and j-th column of the transposed matrix A would be \( a_{ji} \).
Here's what the transpose operation looks like:

• Start with matrix A:
  \[ A = \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \right) \]
• Transpose of A:
  \[ A^T = \begin{pmatrix} 1 & 4 \ 2 & 5 \ 3 & 6 \right) \]

Notice how the rows and columns have been interchanged.
Transpose operations are fundamental in various mathematical transformations and are essential in the study of symmetric and Hermitian matrices.

hermitian matrices

A Hermitian matrix is a complex square matrix that is equal to its own conjugate transpose. This means that for a Hermitian matrix A, the condition is given by:
\[ A = A^* \],
where \( A^* \) represents the conjugate transpose of A.
The conjugate transpose involves taking the transpose of the matrix and then taking the complex conjugate of each element.

• For real entries: A* is simply the transpose A
• For complex entries: Each element in the conjugate transpose is the complex conjugate of the corresponding element in the original matrix.

For example, consider the matrix:
\[ A = \begin{pmatrix} 5 & 2 + i \ 2 - i & 3 \right) \]
The conjugate transpose \( A^* \) is:
\[ A^* = \begin{pmatrix} 5 & 2 - i \ 2 + i & 3 \right) \]
Since \( A = A^* \), it is Hermitian.
Hermitian matrices are crucial in quantum mechanics, signal processing, and complex analysis due to their special properties, such as real eigenvalues.

matrix commutativity

Matrix commutativity is an important concept in linear algebra. Two matrices A and B are said to commute if their product is the same regardless of the order in which they are multiplied. This is mathematically expressed as:
\[ AB = BA \].
In general, matrix multiplication is not commutative, meaning \( AB e BA \) for arbitrary matrices A and B. However, in specific scenarios, like with symmetric or Hermitian matrices, the commutativity condition holds under certain conditions, making the resultant properties (e.g., symmetry or Hermitian property) true for their products.
Understanding when matrices commute is essential for solving many problems in linear algebra, as it can simplify computations and lead to deeper insights into the matrix structure and behavior.