Almaz Khusnutdinov•Math enthusiast, love coding and mathematics
Published 10/4/2025
Unitary Matrices
Computational Problems
Problem 1.1
Let r1=(1+i,2)∈C2.Build a 2×2 unitary U with r1 as its first row by:(a) Normalizing r1.(b) Finding all r2=(x,y) with r1⋅r2=aˉx+bˉy=0.(c) Parametrizing r2 via a phase eiθ.Detailed explanation:(a) ∥r1∥=∣1+i∣2+∣2∣2=2+4=6,r1=61(1+i,2).(b) Solve aˉx+bˉy=0 with a=61+i,b=62.A spanning solution is (−bˉ,aˉ).Indeed, aˉ(−bˉ)+bˉ(aˉ)=−aˉbˉ+bˉaˉ=0.To make r2 unit, take any phase eiθ and set r2=eiθ(−bˉ,aˉ)=eiθ(−62,61−i),which has ∥r2∥=1and r1⋅r2=0.(c) Hence all 2×2 unitaries with first row r1 are U(θ)=[61+i−62eiθ6261−ieiθ],θ∈R.Answer: r1=(61+i,62),r2=eiθ(−62,61−i),U(θ)=[61+i−62eiθ6261−ieiθ].
Problem 1.2
Let r1=(3,−i).(a) Normalize r1.(b) Find all r2=(x,y)withaˉx+bˉy=0.(c) Parametrize with eiθ.Answer: ∥r1∥=10,r1=(103,−10i). All r2=eiθ(−bˉ,aˉ)=eiθ(10i,103).U(θ)=[10310ieiθ−10i103eiθ],θ∈R.
Problem 1.3
Let r1=(2−i,1+2i).(a) Normalize r1.(b) Find all r2 with aˉx+bˉy=0.(c) Parametrize with eiθ.Answer: ∣2−i∣2=5,∣1+2i∣2=5,∥r1∥=10,r1=(102−i,101+2i).r2=eiθ(−bˉ,aˉ)=eiθ(−101−2i,102+i),U(θ)=[102−i−101−2ieiθ101+2i102+ieiθ],θ∈R.
Problem 1.4
Let (a,b)=(0,0).(a) Normalize r1=(a,b).(b) Solve aˉx+bˉy=0 for unit r2.(c) Parametrize all U.Answer: r1=∣a∣2+∣b∣21(a,b). All orthogonal unit rows are r2=eiθ(−bˉ,aˉ)/∣a∣2+∣b∣2.Thus U(θ)=∣a∣2+∣b∣21[a−bˉeiθbaˉeiθ],θ∈R.
Problem 1.5
Let a desired first row be r1=(cosϕ,eiψsinϕ),ϕ∈[0,π/2],ψ∈R.(a) Verify it is unit. (b) Find all r2 with aˉx+bˉy=0.(c)Parametrize U.Answer: ∥r1∥2=cos2ϕ+sin2ϕ=1.r2=eiθ(−bˉ,aˉ)=eiθ(−e−iψsinϕ,cosϕ).U(θ)=[cosϕ−e−iψsinϕeiθeiψsinϕcosϕeiθ],θ∈R.