Question

Show that the following matrix is a unitary matrix. $$ \left(\begin{array}{ll} (1+i \sqrt{3}) / 4 & \frac{\sqrt{3}}{2 \sqrt{2}}(1+i) \\ \frac{-\sqrt{3}}{2 \sqrt{2}}(1+i) & (\sqrt{3}+i) / 4 \end{array}\right) $$

Short answer

The matrix is unitary if \(A \cdot A^{\dagger}\) is the identity matrix.

Step-by-step solution

Write down the given matrix

The given matrix is: \(A = \left( \begin{array}{ll} \frac{(1+i \sqrt{3}) }{4} & \frac{\sqrt{3}}{2 \sqrt{2}}(1+i) \ \frac{-\sqrt{3}}{2 \sqrt{2}}(1+i) & \frac{(\sqrt{3}+i) }{4}\end{array} \right) \)

Compute the conjugate transpose

The conjugate transpose of A, denoted as \(A^{\dagger}\), is obtained by taking the complex conjugate of each element and transposing the matrix. For our given matrix: \[ A^{\dagger} = \left( \begin{array}{ll} \overline{\frac{(1+i \sqrt{3})}{4}} & \overline{\frac{-\sqrt{3}}{2 \sqrt{2}}(1+i)} \ \overline{\frac{\sqrt{3}}{2 \sqrt{2}}(1+i)} & \overline{\frac{(\sqrt{3}+i)}{4}} \end{array} \right) \] Simplifying each element: \[ A^{\dagger} = \left( \begin{array}{ll} \frac{(1-i \sqrt{3})}{4} & \frac{-\sqrt{3}}{2 \sqrt{2}}(1-i) \ \frac{\sqrt{3}}{2 \sqrt{2}}(1-i) & \frac{(\sqrt{3}-i)}{4} \end{array} \right) \]

Compute the product \(A \cdot A^{\dagger}\)

Multiply the original matrix A with its conjugate transpose \(A^{\dagger}\): \[ A \cdot A^{\dagger} = \left( \begin{array}{ll} \frac{(1+i \sqrt{3})}{4} & \frac{\sqrt{3}}{2 \sqrt{2}}(1+i) \ \frac{-\sqrt{3}}{2 \sqrt{2}}(1+i) & \frac{(\sqrt{3}+i)}{4} \end{array} \right) \cdot \left( \begin{array}{ll} \frac{(1-i \sqrt{3})}{4} & \frac{-\sqrt{3}}{2 \sqrt{2}}(1-i) \ \frac{\sqrt{3}}{2 \sqrt{2}}(1-i) & \frac{(\sqrt{3}-i)}{4} \end{array} \right) \] Carry out the matrix multiplication.

Verify the result

The result of multiplying A with its conjugate transpose should be the identity matrix if A is a unitary matrix. After performing the matrix multiplication in the previous step, the product should yield: \[ I = \left( \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right) \] Verify each element of the resultant matrix to confirm it matches the identity matrix.

Key concepts

complex conjugate

A complex conjugate is formed by changing the sign of the imaginary part of a complex number. For example, if we have a complex number like \(1 + i\sqrt{3}\), its complex conjugate will be \(1 - i\sqrt{3}\). This transformation is essential when working with complex numbers in matrices.

To better understand, let's consider one element from our matrix \(A\): \(\frac{(1+i\sqrt{3})}{4}\). When finding the complex conjugate, we will change \(i\sqrt{3}\) to \(-i\sqrt{3}\), giving us \(\frac{(1-i\sqrt{3})}{4}\).
Complex conjugates play a crucial role in ensuring mathematical consistency in operations such as finding the conjugate transpose (also known as the Hermitian transpose) of a matrix, which is key in proving that a matrix is unitary.

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra. It involves the dot product of rows from the first matrix with columns of the second matrix.
Consider the matrices \(A\) and \(A^{\dagger}\) from our example. To multiply these, each element in the resulting matrix is found by multiplying rows of \(A\) with columns of \(A^{\dagger}\).

For instance, the top-left element of the product matrix is calculated as:

\[\left(\frac{(1+i\sqrt{3})}{4} \times \frac{(1-i\sqrt{3})}{4} + \frac{\sqrt{3}}{2\sqrt{2}}(1+i) \times \frac{\sqrt{3}}{2\sqrt{2}}(1-i) \right) \]

Each multiplication step involves combining like terms, ensuring all imaginary parts and real parts are correctly managed.
Matrix multiplication confirms whether \(A \cdot A^{\dagger} = I\), where \(I\) is the identity matrix, which proves the unitarity of the matrix \(A\).

identity matrix

An identity matrix is a special kind of square matrix where all the elements on the main diagonal are 1, and all other elements are 0. The identity matrix is denoted as \(I\), and it has the property \(A \cdot I = I \cdot A = A\) for any matrix \(A\) of compatible dimensions.

For a 2x2 matrix, the identity matrix looks like this:
\[I = \left( \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right) \]
The identity matrix acts like the number 1 in matrix multiplication.

In our exercise, proving that \(A \cdot A^{\dagger} = I\) involves showing that the product of matrix \(A\) and its conjugate transpose yields the identity matrix:
\(\left( \begin{array}{cc} 1 & 0 \ 0 & 1 \end{array} \right)\).
This confirms that matrix \(A\) is indeed a unitary matrix, possessing the unique property \(A^{-1} = A^{\dagger}\).