Question

Show that a matrix A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]` is unitary.

Answer 1

Given A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]`

To prove unitary, we have to prove AA^θ = I

∴ `A^theta = 1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]`

∴ LHS = AA^θ

= `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]`

`=1/4[(2+2+0,-2i+2i+0,0+0+0),(2i-2i+0,2+2+0,0+0+0),(0+0+0,0+0+0,0+0+4)]`

`=1/4[(4,0,0),(0,4,0),(0,0,4)]`

`=[(1,0,0),(0,1,0),(0,0,1)]`

LHS= I
= RHS
LHS =RHS
Hence proved.