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Show that a Matrix a = 1 2 ⎡ ⎢ ⎢ ⎣ √ 2 − I √ 2 0 I √ 2 − √ 2 0 0 0 2 ⎤ ⎥ ⎥ ⎦ is Unitary. - Applied Mathematics 1

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Sum

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

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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.

Concept: Types of Matrices
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