MHB Solving for Unit Vectors in U Given Vector 1 and Matrix S

[cylers89]

So I am given the vector 1...but I need to find vector 2 and 3, in order to find U=(v1, v2, v3), and U is a unitary matrix.

Vector 1 is: (1/2+1/2i 1/2 1/2i)^T

The example from my notes shows me how to find U, but I am also given a matrix S to start with...

Any clue where to start?

 

[Opalg]

So I am given the vector 1...but I need to find vector 2 and 3, in order to find U=(v1, v2, v3), and U is a unitary matrix.

Vector 1 is: (1/2+1/2i 1/2 1/2i)^T

The example from my notes shows me how to find U, but I am also given a matrix S to start with...

Any clue where to start?

A unitary matrix is one whose columns (or rows) form an orthonormal basis for the underlying space. So if the first column v1 is given, then you need to make that the first vector of an orthonormal basis for $\mathbb{C}^3$. The other two vectors v2 and v3 in that basis will then give you the remaining columns of U.

The standard way to construct such a basis is to start with a (non-orthnormal) basis $\{v_1,w_2,w_3\}$ and apply the Gram–Schmidt process to it. You could for example take $w_2=(0,1,0)$ and $w_3=(0,0,1).$

I don't know how the construction in your notes works, but I am guessing that if you take the matrix S to have columns v1, w2 and w3, then that construction will effectively be the same as the Gram–Schmidt construction that I would use.