- #1

[ a, ib; -1 1]

where a and b are constants.

I have to represent and analyse this matrix in a Hilbert space:

I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product:

<x,y> = a*ib -1

and obtain the norm by:

\begin{equation}

||x|| = ia

\end{equation}

to confirm its a unitary matrix in Hilbert space?

However the criteria:

\begin{equation}

<x,y> = a*ib -1 = \overline{<x,y>} = -a*b -i

\end{equation}

is not met.

Instead, if I consider a Hilbert sequence space l^2, then I get:

<x, y> = -ab

where the norm is:

\begin{equation}

||x|| = <x,x>^{1/2} = a

\end{equation}

Is this correct, and if it is, can it be followed up by further analysis of the matrix in Hilbert space?

Thanks!