⚡ Pluto.jl ⚡

Problem 2. Consider the normalized vector in $C^{4}$ :

$v = \frac{1}{2} (\begin{matrix} i \\ - i \\ i \\ - i \end{matrix})$

Show that the matrix $U = I_{4} - 2 v v^{†}$ is unitary and find the corresponding eigenvalues.

10.7 μs

Solution.

15.8 μs

xxxxxxxxxx
 
using SymPy

8.6 s

x
 
begin
    𝑖 = sympy.I
    ⋅(x::T,y) where {T<:Number} = x*y
    ⋅(x::SymMatrix,y) = sympy.MatMul(x.as_explicit(),y,evaluate=true)
    ⋅(x,y::SymMatrix) = sympy.MatMul(x,y.as_explicit(),evaluate=true)
    isunitary(x::Sym) = begin
        sympy.Equality(x.as_explicit().inv(),x.as_explicit().adjoint())
    end
end;

116 μs

x
 
begin
    𝐯 = 1/2 ⋅ [ 𝑖;
               -𝑖;
                𝑖;
                𝑖];
    𝐯ᵈ = sympy.adjoint(𝐯)
    I₄ = sympy.Identity(4) # symbolic identity, could use sympy.eye(4,4)
end;

1.5 ms

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U = I₄ - 2⋅𝐯⋅𝐯ᵈ;

2.7 ms

$U = [\begin{matrix} 0.5 & 0.5 & - 0.5 & - 0.5 \\ 0.5 & 0.5 & 0.5 & 0.5 \\ - 0.5 & 0.5 & 0.5 & - 0.5 \\ - 0.5 & 0.5 & - 0.5 & 0.5 \end{matrix}]$

2.3 ms

Now checking if this matrix is indeed unitary (i.e. $U^{- 1} = U^{†}$ ):

12.4 μs

$True$

x
 
isunitary(U)

7.5 ms

Now finding the eigenvalues:

8.5 μs

x
 
e = U.as_explicit().eigenvects(dict=true,chop=true);

20.0 ms

A total of 4 eigenvalues:

$e_{1} = - 1.0$

$e_{2} = 1.0$

$e_{3} = 1.0$

$e_{4} = 1.0$ .

As one can see 3 of them are degenerate.

4.2 ms