Basic Mathematics For Quantum Computing - Hermitian Matrices

In this post, we’re going to be discussing what Hermitian matrices are.

Transpose of a matrix

Let’s start by learning about the transpose of a matrix. The transpose of a matrix is just flipping the matrix so that every row becomes a column and vice versa.

For example, if I have

\[A = \begin{bmatrix}-5 & 2 \\ -9 & 6\end{bmatrix}\]

Then the transpose is:

\[\begin{bmatrix}-5 & -9 \\ 2 & 6\end{bmatrix}\]

We denote the transpose by writing of a matrix A by writing:

\[A^T\]

If a matrix is its own transpose we may say that the matrix is symmetric

Conjugate transpose of a matrix

If the numbers in a matrix are complex we can also consider the conjugate transpose of the matrix. All this means is that we replace every element by its complex conjugate and then we take the transpose.

So if

\[A = \begin{bmatrix}-5 + 2i & 2+3i \\7+2i & 3\end{bmatrix}\]

Then the conjugate transpose is

\[\begin{bmatrix}-5 - 2i & 7-2i \\ 2 - 3i & 3\end{bmatrix}\]

We denote this by

\[A^\dagger\]

Hermitian matrices

Now we can define Hermitian matrices. A Hermitian matrix is simply one that is equal to its own conjugate transpose so we have:

\[A = A^\dagger\]

For example:

\[\begin{bmatrix}1 & 2+3i \\2-3i & 3\end{bmatrix}\]

is a Hermitian matrix as it is equal to its conjugate transpose.

Unitary matrices

Finally, we can talk about unitary matrices. A matrix, A, is unitary if its inverse is its conjugate transpose.

So we get

\[AA^\dagger = I\]

For example if

\[A = \begin{bmatrix}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\\frac{1}{\sqrt{2}}i & -\frac{1}{\sqrt{2}} i\end{bmatrix}\]

Then

\[A^\dagger = \begin{bmatrix}\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}i \\\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} i\end{bmatrix}\]

So we get

\[AA^\dagger = I\]