First recall that in Chapter 14 we demonstrated (with some effort it must be said) that the generic form of a 2 × 2 Unitary matrix, M, is:

Where a and b ∈ ℂ, θ is an angle with 0 ≤ θ < 2π and |a|² + |b|² = 1.

We can use this to form our modified matrix, M – λI as follows:

We then recall the requirement that the determinant of this matrix is zero:

Again the determinant of a 2 × 2 matrix with entries {a, b, c, d} is simply ad – bc, so we have:

(a – λ)(e^iθ a – λ) + b e^iθ b = 0

Multiplying out, we get:

a e^iθ a – aλ – λ e^iθ a + λ² + b e^iθ b = 0

Regrouping:

λ² – λ(a + e^iθ a) + (a e^iθ a + b e^iθ b) = 0

Or:

λ² – λ(a + a e^iθ) + (a a e^iθ + b b e^iθ) = 0          (1)

From the definition of Unitary matrices, we know that |a|² + |b|² = 1 and from Chapter 14 we can recall that for any complex number, z, we have:

z z = |z|²

This means that:

a a + b b = 1

Or:

b b = 1 – a a

Substituting this in (1) above, we get:

λ² – λ(a + a e^iθ) + (a a e^iθ + (1 – a a) e^iθ) = 0

Multiplying out the bracket again we get:

λ² – λ(a + a e^iθ) + (a a e^iθ + e^iθ – a a e^iθ) = 0

Cancelling the ± a a e^iθ, we get:

λ² – λ(a + a e^iθ) + e^iθ = 0          (2)

Once more in Chapter 14 we also established that for a complex number, z, which forms an angle φ with the x-axis, we have:

z = |z| e^iφ

and

z = |z| e^{– iφ}

If we substitute both a and a with this formulation in (2), we get:

λ² – λ(|a| e^iφ + |a| e^{– iφ} e^iθ) + e^iθ = 0

Which we can simplify to:

λ² – λ|a|(e^iφ + e^{i(θ – φ)}) + e^iθ = 0