1. Vectors
A vector is an ordered list of numbers. In physics, vectors represent positions, velocities, forces, and quantum states.
v = (v₁, v₂, v₃) or |v⟩ = v₁|1⟩ + v₂|2⟩ + v₃|3⟩
Vector Operations
Addition:
(a, b) + (c, d) = (a+c, b+d)
Scalar multiplication:
λ(a, b) = (λa, λb)
Dot product:
u · v = u₁v₁ + u₂v₂ + u₃v₃
Cross product:
u × v = (u₂v₃ - u₃v₂, ...)
In quantum mechanics, we use Dirac notation: |ψ⟩ is a "ket" (column vector), ⟨ψ| is a "bra" (row vector), and ⟨φ|ψ⟩ is the inner product.
2. Matrices
A matrix is a rectangular array of numbers. Matrices represent linear transformations — operations that preserve addition and scalar multiplication.
🎮 Interactive: Matrix Multiplication
A
1
2
3
4
B
5
6
7
8
A×B
19
22
43
50
Hover over result elements to see how they're computed
Important Matrix Types
| Type | Property | Physics Use |
|---|---|---|
| Symmetric | A = Aᵀ | Real observables |
| Hermitian | A = A† | Quantum observables |
| Unitary | U†U = I | Time evolution, rotations |
| Orthogonal | OᵀO = I | 3D rotations |
3. Eigenvalues and Eigenvectors
An eigenvector of matrix A is a special vector that only gets scaled when A acts on it:
A|v⟩ = λ|v⟩
Eigenvalue equation
Here λ is the eigenvalue — the scale factor.
🎮 Interactive: Rotation Matrix and Complex Eigenvalues
Matrix Action: Av
Rotation Matrix
0.87
-0.50
0.50
0.87
Eigenvalues (complex)
λ = 0.87 ± 0.50i
= e±iθ
Rotation matrices have complex eigenvalues (except at 0° and 180°). This is why quantum mechanics needs complex numbers!
Quantum measurements give eigenvalues!
When you measure observable A on state |ψ⟩, you always get one of A's eigenvalues. The eigenvectors are the "definite value" states.
When you measure observable A on state |ψ⟩, you always get one of A's eigenvalues. The eigenvectors are the "definite value" states.
4. The Trace
The trace of a matrix is the sum of its diagonal elements:
Tr(A) = a₁₁ + a₂₂ + ... + aₙₙ = Σᵢ aᵢᵢ
🎮 Interactive: Trace Calculator
0
0
0
0
0
0
Tr(A) =
10
The trace is the sum of diagonal elements: 3 + 5 + 2 = 10
Key property: Tr(AB) = Tr(BA) — the trace is cyclic!
This makes it useful for physics: it doesn't depend on the basis you choose.
This makes it useful for physics: it doesn't depend on the basis you choose.
Trace Properties
Cyclic: Tr(ABC) = Tr(BCA) = Tr(CAB)
Linear: Tr(A + B) = Tr(A) + Tr(B)
Sum of eigenvalues: Tr(A) = λ₁ + λ₂ + ... + λₙ
Z²
rank(G_SM) = Tr(Cartan) = 4
The rank of the Standard Model gauge group equals the trace of its Cartan generators
5. Connection to Z² Framework
Z²
4Z² + 3 = α⁻¹
The '4' comes from rank(SU(3)×SU(2)×U(1)) = 2+1+1 = 4
Z²
Tr(T_a T_b) = δ_ab/2
Generator normalization determines gauge coupling relations
Z²
|ψ|² = ⟨ψ|ψ⟩
Probability from inner product structure
Why Linear Algebra Matters
- • Quantum states are vectors in Hilbert space
- • Observables are Hermitian matrices
- • Symmetries are represented by unitary matrices
- • Gauge groups are matrix Lie groups
Exercises
- Multiply the matrices [[1,2],[3,4]] and [[0,1],[1,0]]. Is AB = BA?
- Find the eigenvalues of [[2,1],[1,2]]. Verify Tr(A) = λ₁ + λ₂.
- Show that rotation matrices satisfy RᵀR = I.
- Calculate Tr(AB) and Tr(BA) for two 2×2 matrices of your choice.
- Why must quantum observables be Hermitian? (Hint: eigenvalues)