Invariants — the properties that survive transformation.
Water molecule in three coordinate systems:
| System | O position | H₁ position | H₂ position |
|---|---|---|---|
| Original | (0, 0, 0) | (0.96, 0, 0) | (-0.24, 0.93, 0) |
| Rotated 90° | (0, 0, 0) | (0, 0.96, 0) | (-0.93, -0.24, 0) |
| Reflected | (0, 0, 0) | (-0.96, 0, 0) | (0.24, 0.93, 0) |
The numbers change completely. But:
Some things don't change. These are invariants.
Transformations alter representations. But certain properties remain fixed — they are intrinsic to the object, not to our description of it.
Finding invariants means finding what's real versus what's coordinate-dependent.
| Situation | Why invariants matter |
|---|---|
| "Is this property physical or just coordinate-dependent?" | Physical = invariant |
| "Are these two matrices really the same transformation?" | Check invariants |
| "What's preserved under this symmetry?" | Find the invariants |
| "Why do similar matrices have the same eigenvalues?" | Eigenvalues are invariants |
| "What quantity is conserved?" | Conservation = invariance under time |
Meaning: A and B represent the same linear transformation in different bases.
Using tr(XY) = tr(YX):
The characteristic polynomial is:
Same polynomial ⟹ same roots ⟹ same eigenvalues.
The invariants capture the essence of the transformation; the rest is coordinate choice.
Enter two matrices and check if they could be similar:
For 2×2 matrices, trace and determinant completely determine the eigenvalues:
| Region | Eigenvalue type |
|---|---|
| tr² - 4det > 0 | Two distinct real |
| tr² - 4det = 0 | One repeated real |
| tr² - 4det < 0 | Complex conjugate pair |
The determinant is the signed volume scaling factor.
| Transformation | det | Meaning |
|---|---|---|
| Rotation | +1 | Volume preserved, orientation preserved |
| Reflection | -1 | Volume preserved, orientation flipped |
| Scaling by k | kⁿ | Volume scales by kⁿ |
| Singular | 0 | Collapses dimension |
The determinant equals the product of all eigenvalues.
The trace equals the sum of all eigenvalues.
Old framing: "Eigenvalues are special scalings along special directions."
New framing: "Eigenvalues are the coordinate-independent quantities that characterize a transformation."
The directions (eigenvectors) depend on your basis. The scalings (eigenvalues) don't.
The eigenvalues are the complete similarity invariants for diagonalizable matrices.
Two diagonalizable matrices are similar if and only if they have the same eigenvalues (with same multiplicities).
In physics, conservation laws are statements of invariance:
| Conservation of... | Invariance under... |
|---|---|
| Energy | Time translation |
| Momentum | Space translation |
| Angular momentum | Rotation |
| Charge | Gauge transformation |
Noether's Theorem: Every continuous symmetry corresponds to a conserved quantity.
Given A and B, are they similar (same transformation, different basis)?
If any differ, not similar.
If all agree, might be similar — need deeper analysis.
But A is not diagonalizable (only one eigenvector) while B = I is diagonal.
Not similar! (Deeper invariant: Jordan structure differs.)
| Transformation type | What's preserved |
|---|---|
| Any invertible P⁻¹AP | det, tr, eigenvalues, rank, char. poly. |
| Orthogonal Q^T AQ | Above + singular values, Frobenius norm |
| Identity (no change) | Everything |
When you see a matrix, ask:
The invariants tell you what's physically meaningful.
Verify that these matrices have the same trace, determinant, and eigenvalues:
Are they similar? If so, find P such that P⁻¹AP = B.
tr(A) = 4 + 1 = 5, tr(B) = 3 + 2 = 5 ✓
det(A) = 4 - (-2) = 6, det(B) = 6 ✓
Eigenvalues of A: λ² - 5λ + 6 = 0 ⟹ λ = 2, 3 ✓ (matches B's diagonal)
Yes, similar. To find P: eigenvectors of A are columns of P.
For λ = 3: (A - 3I)v = 0 ⟹ v₁ = (2, 1)
For λ = 2: (A - 2I)v = 0 ⟹ v₂ = (1, 1)
P = [2 1; 1 1], verify P⁻¹AP = B.
Show that these matrices are not similar, despite having the same trace and determinant:
Both have tr = 4, det = 4, eigenvalue λ = 2 (double).
But A has only one eigenvector direction: (A - 2I) = [0 1; 0 0], null space = span{(1,0)}.
B = 2I has every vector as eigenvector (full eigenspace).
Different geometric multiplicity ⟹ not similar.
For each 2×2 matrix, use trace and determinant to classify the eigenvalues:
(a) $\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$
(b) $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$
(c) $\begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$
(a) tr = 6, det = 9 - 1 = 8. Discriminant = 36 - 32 = 4 > 0. Two distinct real.
(b) tr = 0, det = 1. Discriminant = 0 - 4 = -4 < 0. Complex conjugate pair (±i).
(c) tr = 4, det = 4. Discriminant = 16 - 16 = 0. Repeated real (λ = 2).
In the reaction 2H₂ + O₂ → 2H₂O:
(a) What quantities are conserved (invariant)?
(b) What quantities change?
(a) Conserved: Total mass, total charge (0), atom counts (4 H, 2 O), total energy (in isolated system), total momentum.
(b) Changes: Number of molecules (3 → 2), bond types, molecular geometries, potential energy distribution.
Lecture 10: What Survives?
Now that we know eigenvalues are fundamental invariants, how do we find them? The characteristic equation, eigenvectors, and applications to chemistry.
Chemical Thinking: The Grammar of Reality
Lecture 9 — What Doesn't Change?