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Question
Show that a matrix A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]` is unitary.
Solution
Given A = `1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]`
To prove unitary, we have to prove AAθ = I
∴ `A^theta = 1/2[(sqrt2,-isqrt2,0),(isqrt2,-sqrt2,0),(0,0,2)]`
∴ LHS = AAθ
= `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.
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