Show that the definition of a Hermitian matrix (A \(=\mathrm{A}^{\dagger}\) ) can be written \(a_{i j}=\bar{a}_{j i}\) (that is, the diagonal elements are real and the other elements have the property that \(\left.a_{12}=\bar{a}_{21}, \text { etc. }\right) .\) Construct an example of a Hermitian matrix.

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 \(A^{\dagger}\) 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 \(A^{\dagger}\) 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: \(A = A^{\dagger}\). This means for every element \(a_{ij}\), it should hold that \(a_{ij} = \bar{a}_{ji}\).

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 \(\bar{z} = a - bi\). 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 \(a_{ij} = \bar{a}_{ji}\). 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.

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\(a_{11} = \bar{a}_{11}\). 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: \[A = \begin{pmatrix} a & b + ic & d + ie \ b - ic & f & g + ih \ d - ie & g - ih & k \end{pmatrix}\]
\(a, f,\) 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.