How to convert a random matrix to Unitary Matrix?

Question

I know that a complex matrix $n \times n$ is said to be unitary if $AA^*=A^*A=I$ or equivalently if $A^*=A^{-1}$. But I asked what if there is a random matrix and we want to turn it into an unitary matrix, please also give an example.

Answer 1

If you have access to Matlab or Octave, then producing such matrices is as easy as issuing the following commands:

n = 10;
A = rand(n) + 1i*rand(n);
[U,~] = qr(A);

I recommend that you use software such as this to generate these matrices in general. For small matrices that you would like to work by hand, the Gram-Schmidt procedure is what you want. Here is a simple example involving a real $3\times 3$ matrix. Let $$ A = \begin{bmatrix} 1 & 1 & 1\\ 1 & 0 & 2\\ 0 & 2 & 1 \end{bmatrix} $$
We are going to construct a $3\times 3$ unitary matrix $U$ from the columns of $A$. For simplicity I will first construct a matrix $\hat{U}$ with orthogonal columns, and then normalize at the end to get a unitary matrix. The first step is easy, we set the first column of $\hat{U}$ equal to the first column of $A$, i.e., $\hat{u}_1 = a_1$. The second column of $\hat{U}$ is equal to $a_2$ minus any contributions from $\hat{u}_1$, that is, $$ \hat{u}_2 = a_2 - \frac{\hat{u}_1\cdot a_2}{\hat{u}_1\cdot \hat{u}_1}\hat{u}_1 = \begin{bmatrix} 1\\ 0\\ 2 \end{bmatrix} - \frac{1}{2} \begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix} = \begin{bmatrix} \phantom{-}1/2\\ -1/2\\ \phantom{-}2 \end{bmatrix} $$ The last step is the same as the second except now we must remove from $a_3$ any contributions from $\hat{u}_1$ or $\hat{u}_2$. Thus, \begin{align} \hat{u}_3 &= a_3 - \frac{\hat{u}_1\cdot a_3}{\hat{u}_1\cdot \hat{u}_1}\hat{u}_1 - \frac{\hat{u}_2\cdot a_3}{\hat{u}_2\cdot \hat{u}_2}\hat{u}_2\\[1mm] &= \begin{bmatrix} 1\\ 2\\ 1 \end{bmatrix} - \frac{3}{2} \begin{bmatrix} 1\\ 1\\ 0 \end{bmatrix} - \frac{1}{3} \begin{bmatrix} \phantom{-}1/2\\ -1/2\\ \phantom{-}2 \end{bmatrix}\\[1mm] &= \begin{bmatrix} -2/3\\ \phantom{-}2/3\\ \phantom{-}1/3 \end{bmatrix} \end{align} The last step to obtain $U$ from $\hat{U}$ is to normalize the columns of $\hat{U}$. Doing so we obtain the unitary matrix $$ U = \begin{bmatrix} 1/\sqrt{2} & \phantom{-}1/(3\sqrt{2}) & -2/3\\ 1/\sqrt{2} & -1/(3\sqrt{2}) &\phantom{-}2/3\\ 0 & \phantom{-}2\sqrt{2}/3 &\phantom{-}1/3 \end{bmatrix} $$ As you can see from this simple example, the procedure is quite tedious by hand and is best left to a computer. You can read more about the Gram-Schmidt procedure here.