Question

Show that the definition of a Hermitian matrix (A $={\mathrm{A}}^{†}$ ) can be written $a_{ij}={\overline{a}}_{ji}$ (that is, the diagonal elements are real and the other elements have the property that $a_{12}={\overline{a}}_{21},\text{ etc. }).$ Construct an example of a Hermitian matrix.

Short answer

The Hermitian matrix definition $A=A^{†}$ means $a_{ij}={\overline{a}}_{ji}$. Example: $\left(\begin{array}{ccccccc}a& b+ic& d+ie\text{b}-ic& f& g+ih\text{d}-ie& g-ih& k\end{array}\right)$.

Step-by-step solution

- Understand the Hermitian Matrix Definition

A matrix $A$ is Hermitian if it is equal to its own conjugate transpose, i.e., $A=A^{†}$. The conjugate transpose $A^{†}$ is obtained by taking the transpose of $A$ and then taking the complex conjugate of each element.

- Express the Elements of the Conjugate Transpose

Let $A=[a_{ij}]$. The element in the $i$-th row and $j$-th column of $A^{†}$ is ${\overline{a}}_{ji}$, where the bar denotes the complex conjugate.

- Compare Elements from the Hermitian Definition

For $A$ to be Hermitian, $a_{ij}={\overline{a}}_{ji}$ for all elements. This means that elements along the diagonal $a_{ii}$ must be equal to their own complex conjugate, implying they are real. For off-diagonal elements, $a_{12}={\overline{a}}_{21}$, $a_{13}={\overline{a}}_{31}$, etc.

- Construct an Example Matrix

Consider the matrix: $$A=\left(\begin{array}{ccccccc}a& b+ic& d+ie\text{b}-ic& f& g+ih\text{d}-ie& g-ih& k\end{array}\right)$$ - Here, $a,f,k$ are real since they must be equal to their own complex conjugates. - The off-diagonal elements like $b+ic$ and $d+ie$ are such that $a_{12}={\overline{a}}_{21}$, $a_{13}={\overline{a}}_{31}$, etc.

Key concepts

Conjugate Transpose

To understand a Hermitian matrix, it's important to first grasp the concept of a conjugate transpose. The conjugate transpose of a matrix is denoted as $A^{†}$ and involves two operations:

• Taking the transpose of the matrix.
• Taking the complex conjugate of each element in the transposed matrix.

Taking the transpose means swapping rows with columns. For instance, the element at the $i$-th row and $j$-th column ($a_{ij}$) in the original matrix will now be positioned at the $j$-th row and $i$-th column in the transposed matrix ($a_{ji}$).
The second operation, taking the complex conjugate, involves changing the sign of the imaginary part of each complex number. For example, if an element is $2+3i$, its complex conjugate is $2-3i$. Thus, the conjugate transpose $A^{†}$ of matrix $A$ is the matrix you get after applying both these operations.

The definition of a Hermitian matrix states that it must be equal to its conjugate transpose: $A=A^{†}$. This means for every element $a_{ij}$, it should hold that $a_{ij}={\overline{a}}_{ji}$.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. If you have a complex number $z=a+bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, the complex conjugate of $z$ is $\overline{z}=a-bi$. The operation of taking a complex conjugate is important to understand Hermitian matrices.

Hermitian matrices use this concept by requiring that each element and its corresponding element from the conjugate transpose satisfy the relationship $a_{ij}={\overline{a}}_{ji}$. How does this look in practice? Let's say you have a matrix element $a_{12}$, which is $3+4i$. For the matrix to be Hermitian, the element $a_{21}$ needs to be its complex conjugate, which means $a_{21}=3-4i$.

This means for every non-diagonal element, you need to check its counterpart in the conjugate transpose to ensure they are complex conjugates of each other. This property ensures the 'mirror-like' symmetry Hermitian matrices are known for.

Diagonal Elements

In a Hermitian matrix, diagonal elements (the ones indexing the same row and column, like $a_{11},a_{22},...a_{nn}$) have a special property: they must be real numbers. This comes from the requirement that each diagonal element must be equal to its own complex conjugate. A number is equal to its complex conjugate only if it has no imaginary part, meaning it is real.

For example, consider the matrix element $a_{11}$. Using the definition of a Hermitian matrix, we have$a_{11}={\overline{a}}_{11}$. If $a_{11}$ was $2+3i$, its complex conjugate would be $2-3i$, which is not equal to $a_{11}$. Thus, for the Hermitian property to hold, $a_{11}$ must be just $2$, making it a real number.

Constructing a Hermitian matrix then involves ensuring all diagonal elements are real. For example, in the sample matrix: $$A=\left(\begin{array}{ccccccc}a& b+ic& d+ieb-ic& f& g+ihd-ie& g-ih& k\end{array}\right)$$
$a,f,$ and $k$ are real numbers. Ensuring these elements have no imaginary parts upholds the Hermitian property. Thus, by remembering this rule about diagonal elements, constructing and verifying Hermitian matrices becomes simpler.