From wikipedia page https://en.wikipedia.org/wiki/Unitary_matrix Section: Elementary constructions.
What I understand about Unitary matrix is : If we have a square matrix (say 2×2) with complex values. We can say it is Unitary matrix if its transposed conjugate is same of its inverse.
One example is provided in the above mentioned page, where it says it depends on 4 parameters:
- The phase of a,
- The phase of b,
- The relative magnitude between a and b, and
- the angle φ
In one of its example it also says: $|a^2|+|b^2|=1$
Question 1, is it also a property of the U-matrix that $|a^2|+|b^2|=1$
Question 2, if the answer of question 1 is yes, then how should we compute the value of a and b (complex numbers) for output 1. The only option that I know of is taking $a = cos(\theta)$ and $b = sin(\theta)$. But then it is not a complex number.
Question 3, If my assumption of $a = cos(\theta)$ and $b = sin(\theta)$ is correct, then how we calculate phase of a and phase of b. Since it is now a real number.
Best Answer
As in the wiki page, take $$ U = \pmatrix{a&b\\e^{-i\varphi}b^* & e^{i\varphi}a^*}. $$
To see that we must have $|a|^2 + |b|^2 = 1$, it suffices to consider the $(1,1)$ entry of $UU^\dagger$. We compute $$ U^\dagger = \pmatrix{a^*&e^{i\varphi} b\\b^* & e^{-i\varphi }a}, \quad UU^\dagger = \pmatrix{aa^* + bb^* & ?\\? & ?}. $$ However, we must have $UU^\dagger = I$, which means that it must hold that $aa^* + bb^* = 1$. In other words, $|a|^2 + |b|^2 = 1$.
More generally, we can show that every row and every column of a unitary matrix must have entries whose magnitudes sum to $1$. This is briefly mentioned on the wiki page in conditions 4 and 5 from the list of "equivalent conditions".
Regarding your second question: every pair of numbers $a,b$ satisfying $|a|^2 + |b|^2 = 1$ will have the form $$ a = e^{i \psi_1}\cos\theta , \quad b = e^{i\psi_2}\sin \theta $$ for some (real) angles $\theta,\psi_1,\psi_2$. The phase of $a$ is $\psi_1$, and the phase of $b$ is $\psi_2$.