Symmetric and Hermitian Matrices

1. Symmetric Matrices: The Real Case

Let's start in the familiar territory of real numbers. You've encountered the transpose of a matrix, $A^T$, which is formed by flipping the matrix across its main diagonal. A symmetric matrix is simply a matrix that is its own transpose.

Definition: Symmetric Matrix
A square matrix $A$ with real entries is symmetric if it is equal to its transpose:
$A=A^T$
This means that for any row index $i$ and column index $j$, the element $A_{ij}$ is equal to the element $A_{ji}$.

For example, this matrix is symmetric:

To formalize this and see a list of key properties, please read the first section of the following resource.

Symmetric matrices are special for two incredible reasons that we will prove in later lessons:

1. All of their eigenvalues are real numbers.
2. They always have a full set of orthogonal eigenvectors, which means they are always diagonalizable.

To get a first look at these remarkable properties and why they are so important, let's watch the beginning of a classic lecture by Prof. Gilbert Strang.

2. Generalizing to Complex Numbers: The Conjugate Transpose

Your goal of studying quantum mechanics requires us to work with complex numbers. If we simply apply the definition of symmetry ($A=A^T$) to complex matrices, we unfortunately lose the guaranteed real eigenvalues. We need a new definition that preserves this crucial property.

This "correct" generalization involves a new operation: the conjugate transpose. This operation, as its name suggests, involves two steps:

1. Complex Conjugate: Take the complex conjugate of every entry in the matrix. The conjugate of a number $a+bi$ is $a-bi$.
2. Transpose: Take the transpose of the resulting matrix.

This combined operation is also called the Hermitian conjugate or adjoint, and is denoted by a dagger symbol ($†$) or a superscript $H$. So, $A^{†}=(\bar{A}{)}^T$.

The following video provides an excellent step-by-step guide through complex matrices and the definition of the conjugate transpose.

3. Hermitian Matrices: The Complex Analogue of Symmetry

Now that we have the conjugate transpose operation, we can define the complex analogue of a symmetric matrix.

Definition: Hermitian Matrix
A square matrix $A$ with complex entries is Hermitian if it is equal to its own conjugate transpose:
$A=A^{†}$
This means that for any row index $i$ and column index $j$, the element $A_{ij}$ is equal to the complex conjugate of the element $A_{ji}$, or $A_{ij}=\overline{A_{ji}}$.

Notice that if a matrix $A$ contains only real numbers, the conjugation step does nothing ($\bar{A}=A$), so $A^{†}=A^T$. This means a real Hermitian matrix is simply a symmetric matrix. The Hermitian property is the more general concept.

Let's continue with the video to see this definition in action.

The image below gives a perfect summary of the definition and an important consequence.

To solidify this, let's return to our text resource.

Just like symmetric matrices, Hermitian matrices are guaranteed to have real eigenvalues and a full set of orthogonal eigenvectors, making them central to the mathematics of quantum mechanics.

Conclusion

In this lesson, we have defined two exceptionally important types of matrices that form the bedrock for many advanced applications.

Key Takeaways:

• A symmetric matrix is a real matrix that equals its own transpose ($A=A^T$).
• The conjugate transpose ($A^{†}$) of a matrix is found by taking the transpose and then the complex conjugate of every element.
• A Hermitian matrix is a complex matrix that equals its own conjugate transpose ($A=A^{†}$).
• The Hermitian property is a generalization of the symmetric property to complex spaces. A real Hermitian matrix is a symmetric matrix.
• Key properties of Hermitian matrices include having real diagonal elements. More importantly (as we will see), they always have real eigenvalues.

Preview of the Next Lesson:

We've stated that symmetric and Hermitian matrices have real eigenvalues. In the next lesson, we will prove this fundamental property. The proof itself is elegant and will make use of the conjugate transpose operation we learned today, further highlighting its importance for working in complex vector spaces. This will be our first major step in building the mathematical framework required for quantum mechanics.