Component is diagonal in the basis

[vredina97]

Hi, I am analyzing the paper for my thesis and have

Equation

IF=-1*(1+cos(δ))*cot(θ)*σ_2-sin(δ)*cot(θ)*σ_1

where σ_1={{0,1},{1,0}} and σ_2={{0,-i},{i,0}} are the Pauli matrices

The component σ_1 is diagonal in the (1/√2, +/- 1/√2) linear polarization basis and the component σ_2 is diagonal in the circular polarization basis (1/√2, +/- i/√2).

What is mathematical representation of σ_1 and σ_2 being diagonal in that basis.

Thanks in advance,

vredina97

 

[HallsofIvy]

What is mathematical representation of σ_1 and σ_2 being diagonal in that basis.

What do you mean by "that basis"? You refer to two bases, the "linear polarization basis" and the "circular polarization basis". Are you referring to one of those or some other basis?

 

[vredina97]

The σ_1={{0,1},{1,0}} and σ_2={{0,-i},{i,0}} are the Pauli matrices

I have to find diagonal to σ_1 in the (1/√2, +/- 1/√2) linear polarization basis and diagonal to σ_2 in the circular polarization basis (1/√2, +/- i/√2).

I found that diagonal to σ_1 is {{1,0},{0,-1}} (where 1 and -1 are eigenvalues of σ_1={{0,1},{1,0}}) in {1/√2}{1/√2} and {1/√2}{- 1/√2} basis.

But I have troubles with σ_2={{0,-i},{i,0}} where i get eigenvalues lambda=√-1=√i^2. If eigenvalue is complex then matrix can not be diagonalized? or is there another solution?

Thanks in advance,

vredina97