Definition (Complex conjugate of a matrix): The matrix derived by taking the complex conjugates of all the entries of the matrix.
Example: Let \(A=\begin{bmatrix} 1+i & 2-3i\\ 2\cdot e^{i\frac{\pi}{5}} & -10 \end{bmatrix}\), then the complex conjugate of A is $$\bar A= \begin{bmatrix} 1-i& 2+3i\\ 2\cdot e^{-i\frac{\pi}{5}} & -10 \end{bmatrix}.$$

Definition (Transpose of a matrix):
For a square matrix \(A\), the element on the \(i^{th}\) row and \(j^{th}\) column is denoted by \(a_{ij}\). (e.g. for the above example, \(a_{12}= 2-3i\) while \(a_{21}=2\cdot e^{i\frac{\pi}{5}}\).)
Then, for the transpose of \(A\), denoted by \(A^T\), the the element on the \(i^{th}\) row and \(j^{th}\) column would be \(a_{ji}\).
Example: Let \(A\) be same as above. Then \(A^T\) is: \(\begin{bmatrix} 1+i & 2\cdot e^{i\frac{\pi}{5}} \\ 2-3i & -10 \end{bmatrix}\).

Definition (Conjugate Transpose of a matrix):
The conjugate transpose of a square matrix \(A\) is \(A^\dagger := (\bar A)^T\).

Definition (Unitary matrices):
A square matrix is a matrix that has the same number of rows and columns.
Any square matrix \(U\) that satisfies \(U U^\dagger=U^\dagger U= I\) is a unitary matrix.

For example, for the matrix \(Z = \begin{bmatrix}
4-i & 0 \\
0 & 4 + i
\end{bmatrix}\),

we can calculate the unitary matrix by first getting
\(Z^\dagger = \begin{bmatrix}
4+i & 0 \\
0 & 4-i
\end{bmatrix}\) and then multiplying the two together to get \(ZZ^\dagger = \begin{bmatrix}
17 & 0 \\
0 & 17
\end{bmatrix}\).

But, this isn’t \(I = \begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}\) so we need to multiply \(Z\) by \(\frac{1}{\sqrt{17}}\).

Thus, the unitary matrix would be \(U = \frac{1}{\sqrt{17}}\begin{bmatrix}
4-i & 0 \\
0 & 4+i
\end{bmatrix}\).

But, not all matrices can be made unitary matrices. All unitary matrices have certain conditions on them; for example, in order for a matrix to be unitary, it’s rows and columns mod squared all have to be equal to 1. In the matrix \(U=\begin{bmatrix}a&b\\c&d\end{bmatrix}\): \(|a|^2 + |b|^2\), \(|c|^2 + |d|^2\), \(|a|^2 + |c|^2\), and \(|b|^2 + |d|^2\) all have to be equal to one. Keep in mind, modulus squared means just the coefficients of a+bi, so just \(a^2+b^2\). For 1+i, \(|1+i|^2 = 2\).

Also, from this it is obvious that some cannot be made into unitary matrices such as \(A=\begin{bmatrix}1&0\\1&0\end{bmatrix}\).

Practice problem: Find a and b so that \(U = \begin{bmatrix}
cos(\frac{\theta}{2}) & a \\
e^{i\phi}sin(\frac{\theta}{2}) & b
\end{bmatrix}\) and \(UU^\dagger = I\).