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The Future of Now · UChicago QSNC companion
⚙️ The Stretch Machine
MODULE 1 · EXERCISE 2
Finding a gate's special "stretch numbers" — the eigenvalues
🎯 The big idea (for everyone)
A 2×2 gate is a machine that grabs arrows and stretches or spins them.
Most arrows get twisted to a new direction. But a few special arrows keep pointing the same way —
they only get longer or shorter. Those length-changes are the eigenvalues (the "stretch numbers").
First a safety check: if the machine's determinant is not zero,
the only arrow it can squash to nothing is the zero arrow. That means the machine is reversible — you can always undo it.
Executive / UChicago layer: For N = [[a,b],[c,d]] on ℂ², Step 1 shows
det N ≠ 0 ⟹ (Nv = 0 ⟹ v = 0), so N is invertible (if det N = 0, v = (q,−p)ᵀ solves Nv = 0).
Step 2: eigenvalues solve the characteristic equation
λ² − (tr N)λ + det N = 0, giving
λ = ½[(a+d) ± √((a+d)² − 4(ad−bc))]. Note λ₁λ₂ = det N and λ₁+λ₂ = tr N.
🎮 Build your own machine
Move the four dials to set the machine. The faint shape shows how it stretches
a ring of arrows. The gold arrows are the special ones that don't turn — their lengths are the eigenvalues.
trace (a+d)5
det (ad−bc)6
reversible?YES
λ₁3.00
λ₂2.00
λ₁·λ₂ = det?✓
🧠 Quick quiz — earn your stars
1) An eigenvalue (stretch number) tells you…
2) If the determinant is not zero, the machine is…
3) The two stretch numbers always multiply to give…
🏆 Nailed it! You just solved an eigenvalue problem on ℂ² — and proved invertibility from the determinant.