Lecture 10: What Survives?

Eigenvalues and eigenvectors — the directions that transformation preserves.

The Hook

Apply this matrix to many vectors:

$$A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$$

Most vectors change direction. But watch these two:

v₁ = (1, 1): Av₁ = (3, 3) = 3(1, 1) = 3v₁
v₂ = (1, -1): Av₂ = (1, -1) = 1(1, -1) = 1v₂

These vectors only get scaled — their direction survives!

v₁ is scaled by 3. v₂ is scaled by 1 (unchanged).

These special directions are eigenvectors. The scale factors are eigenvalues.

Eigenvector Visualizer

a: 2

b: 1

c: 1

d: 2

Trace (a + d)

Det (ad - bc)

λ₁

λ₂

Recognition: The SAMENESS Primitive

SAMENESS: "What doesn't change?"

Under a linear transformation, most directions rotate. But some directions are special — they only stretch or compress, never rotate.

Finding these invariant directions reveals the essential structure of the transformation.

When to Use Eigenvalues/Eigenvectors

Situation	Eigenanalysis gives you
"What directions does this matrix preserve?"	Eigenvectors
"By how much are they scaled?"	Eigenvalues
"Simplify this matrix?"	Diagonalization
"Solve coupled differential equations?"	Decouple via eigenvectors
"Find normal modes of vibration?"	Eigenvectors of Hessian
"Find molecular orbitals?"	Eigenvectors of Hamiltonian
"Long-term behavior of iterated process?"	Dominant eigenvalue
"Is this matrix stable?"	Check if all \|λ\| < 1
"Principal components?"	Eigenvectors of covariance

Definition

Eigenvector and Eigenvalue

Definition: Let A be an n × n matrix. A nonzero vector v is an eigenvector of A if: $$A\mathbf{v} = \lambda \mathbf{v}$$ for some scalar λ. The scalar λ is the eigenvalue corresponding to v.

Interpretation:

Av is parallel to v (same or opposite direction)
λ > 1: eigenvector is stretched
0 < λ < 1: eigenvector is compressed
λ < 0: eigenvector is flipped and scaled
λ = 1: eigenvector is unchanged
λ = 0: eigenvector is sent to zero (in the kernel)

Terminology:

"eigen" = German for "own" or "characteristic"
Eigenvalue also called: characteristic value, proper value, latent root
The set of all eigenvalues is the spectrum of A

The Eigenspace

Definition: For eigenvalue λ, the eigenspace E_λ is: $$E_\lambda = \{\mathbf{v} : A\mathbf{v} = \lambda\mathbf{v}\} = \ker(A - \lambda I)$$

This is a subspace containing all eigenvectors for λ (plus the zero vector).

Finding Eigenvalues

The Characteristic Equation

Key insight: Av = λv can be rewritten:

$$A\mathbf{v} - \lambda\mathbf{v} = \mathbf{0}$$ $$(A - \lambda I)\mathbf{v} = \mathbf{0}$$

For a nonzero solution v to exist, (A - λI) must be singular:

$$\det(A - \lambda I) = 0$$

This is the characteristic equation.

The Characteristic Polynomial

Definition: The characteristic polynomial of A is: $$p(\lambda) = \det(A - \lambda I)$$

The eigenvalues are the roots of p(λ) = 0.

2×2 Case

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$:

$$A - \lambda I = \begin{pmatrix} a - \lambda & b \\ c & d - \lambda \end{pmatrix}$$

$$p(\lambda) = (a - \lambda)(d - \lambda) - bc = \lambda^2 - (a+d)\lambda + (ad - bc)$$

Key formula: $$p(\lambda) = \lambda^2 - \text{tr}(A)\lambda + \det(A)$$

The eigenvalues satisfy:

Sum: λ₁ + λ₂ = tr(A)
Product: λ₁ · λ₂ = det(A)

Quadratic formula:

$$\lambda = \frac{\text{tr}(A) \pm \sqrt{\text{tr}(A)^2 - 4\det(A)}}{2}$$

Example: $A = \begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$

tr(A) = 4, det(A) = 4 - 1 = 3

$$\lambda = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm 2}{2}$$

λ₁ = 3, λ₂ = 1

Check: 3 + 1 = 4 = tr(A) ✓, 3 × 1 = 3 = det(A) ✓

Special Cases

Triangular matrices: Eigenvalues are the diagonal entries.

$$\begin{pmatrix} a & * & * \\ 0 & b & * \\ 0 & 0 & c \end{pmatrix} \Rightarrow \lambda_1 = a, \lambda_2 = b, \lambda_3 = c$$

Finding Eigenvectors

Once you have eigenvalue λ, find eigenvectors by solving:

$$(A - \lambda I)\mathbf{v} = \mathbf{0}$$

This is a homogeneous system — find the null space of (A - λI).

Example continued: A = [2 1; 1 2], λ₁ = 3

$$A - 3I = \begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}$$

Solve: -v₁ + v₂ = 0, so v₁ = v₂.

Eigenvector: v₁ = (1, 1) (or any scalar multiple)

For λ₂ = 1:

$$A - I = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$$

Solve: v₁ + v₂ = 0, so v₂ = -v₁.

Eigenvector: v₂ = (1, -1)

Step-by-Step Eigenvalue Calculator

Properties of Eigenvalues

Algebraic and Geometric Multiplicity

Algebraic multiplicity: The multiplicity of λ as a root of the characteristic polynomial.
Geometric multiplicity: dim(E_λ) = dimension of the eigenspace.
Always: geometric ≤ algebraic.

Key Properties

A matrix and its transpose have the same eigenvalues.
Similar matrices have the same eigenvalues: if B = P⁻¹AP, then B and A have the same spectrum.
tr(A) = sum of eigenvalues (counting multiplicity).
det(A) = product of eigenvalues.
A is invertible ⟺ all eigenvalues are nonzero.
Eigenvalues of A⁻¹ are 1/λᵢ (with same eigenvectors).
Eigenvalues of Aᵏ are λᵢᵏ (with same eigenvectors).

Special Matrix Types

Symmetric matrices (A = Aᵀ):

All eigenvalues are real
Eigenvectors for distinct eigenvalues are orthogonal
Always diagonalizable

Physical matrices (Hamiltonians, Hessians, covariance matrices) are typically symmetric — the spectral theorem guarantees nice behavior.

Orthogonal matrices (AᵀA = I):

All eigenvalues have |λ| = 1
Eigenvalues are ±1 or come in complex conjugate pairs on unit circle

Positive definite matrices:

All eigenvalues are positive
vᵀAv > 0 for all nonzero v

Complex Eigenvalues

When They Occur

Real matrices can have complex eigenvalues. This happens when:

$$\text{tr}(A)^2 - 4\det(A) < 0$$

The eigenvalues are complex conjugates: λ = α ± βi.

Example: Rotation by θ ≠ 0°, 180°

$$R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$$

tr(R) = 2cos θ, det(R) = 1

$$\lambda = \frac{2\cos\theta \pm \sqrt{4\cos^2\theta - 4}}{2} = \cos\theta \pm i\sin\theta = e^{\pm i\theta}$$

The eigenvalues are complex! (No real direction is preserved by a rotation.)

Complex Eigenvalues: Rotation

Rotation angle: 60°

λ₁

e^(i60°)

λ₂

e^(-i60°)

|λ|

Interpretation

Complex eigenvalues indicate rotation within a plane.

|λ| = 1: pure rotation (no scaling)
|λ| > 1: spiral outward
|λ| < 1: spiral inward

Diagonalization

The Goal

Diagonalization means finding P such that:

$$P^{-1}AP = D = \begin{pmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{pmatrix}$$

The Method

If A has n linearly independent eigenvectors v₁, ..., vₙ:

Form P with eigenvectors as columns: P = [v₁ | v₂ | ... | vₙ]
Then P⁻¹AP = D where D has eigenvalues on diagonal

Why it works:

$$AP = A[\mathbf{v}_1 | \cdots | \mathbf{v}_n] = [A\mathbf{v}_1 | \cdots | A\mathbf{v}_n] = [\lambda_1\mathbf{v}_1 | \cdots | \lambda_n\mathbf{v}_n]$$

$$= [\mathbf{v}_1 | \cdots | \mathbf{v}_n] \begin{pmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \end{pmatrix} = PD$$

Therefore: AP = PD ⟹ P⁻¹AP = D

When Diagonalization Fails

A matrix is not diagonalizable if it doesn't have n linearly independent eigenvectors.

This happens when geometric multiplicity < algebraic multiplicity for some eigenvalue.

Example of non-diagonalizable matrix:

$$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$$

Characteristic polynomial: (1-λ)² = 0, so λ = 1 (algebraic multiplicity 2).

Eigenspace: A - I = [0 1; 0 0]. Null space is span{(1, 0)}. Geometric multiplicity = 1.

Only one independent eigenvector — not diagonalizable.

Why Diagonalization Matters

In the eigenvector basis, the transformation is just scaling along axes:

$$D = P^{-1}AP \Rightarrow A = PDP^{-1}$$

Powers: $A^k = PD^kP^{-1}$ (and Dᵏ is trivial — just raise diagonal entries to kth power)

Exponentials: $e^{At} = Pe^{Dt}P^{-1}$ (useful for differential equations)

Chemistry Connection: Molecular Orbitals

The Secular Equation

In Hückel theory, molecular orbital energies are eigenvalues of the Hamiltonian matrix:

$$\mathbf{H}\mathbf{c} = E\mathbf{c}$$

where:

H is the Hückel matrix
c is the vector of MO coefficients
E is the orbital energy (eigenvalue)

Hückel Molecular Orbital Calculator

Molecule:

Example: Ethylene

$$\mathbf{H} = \begin{pmatrix} \alpha & \beta \\ \beta & \alpha \end{pmatrix}$$

Setting x = (α - E)/β:

$$\det\begin{pmatrix} \alpha - E & \beta \\ \beta & \alpha - E \end{pmatrix} = (\alpha - E)^2 - \beta^2 = 0$$

(α - E) = ±β

E₁ = α + β (bonding), E₂ = α - β (antibonding)

Since β < 0, E₁ < E₂.

Eigenvectors (MO coefficients):

For E₁ = α + β: c₁ = (1, 1)/√2 (bonding: in-phase)
For E₂ = α - β: c₂ = (1, -1)/√2 (antibonding: out-of-phase)

Chemistry Connection: Normal Modes

Vibrational Analysis

The Hessian matrix of second derivatives:

$$H_{ij} = \frac{\partial^2 V}{\partial q_i \partial q_j}$$

Eigenvalue problem: Hq = ω²q

Eigenvalues ω² give vibrational frequencies
Eigenvectors give normal mode displacement patterns
Each mode oscillates independently

Chemistry Connection: Quantum Mechanics

The Eigenvalue Problem

The time-independent Schrödinger equation:

$$\hat{H}\psi = E\psi$$

This IS an eigenvalue problem:

Operator Ĥ (Hamiltonian) → "matrix" (infinite-dimensional)
ψ (wavefunction) → eigenvector
E (energy) → eigenvalue

Allowed energies are eigenvalues. Stationary states are eigenvectors.

Summary: Eigenvalue Toolkit

Step	Method
1. Find eigenvalues	Solve det(A - λI) = 0
2. For each λ, find eigenvectors	Solve (A - λI)v = 0
3. Check	Verify Av = λv

Quick Reference

Property	Formula
Characteristic polynomial	p(λ) = det(A - λI)
Sum of eigenvalues	tr(A)
Product of eigenvalues	det(A)
Eigenvalues of Aᵏ	λᵢᵏ
Eigenvalues of A⁻¹	1/λᵢ
Symmetric ⟹	Real eigenvalues, orthogonal eigenvectors

Exercises

Exercise 1: 2×2 Eigenvalues

Find the eigenvalues and eigenvectors of:

(a) $\begin{pmatrix} 4 & 2 \\ 1 & 3 \end{pmatrix}$

(b) $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

Show Solution

(a) tr = 7, det = 12 - 2 = 10. λ = (7 ± √(49-40))/2 = (7 ± 3)/2. λ₁ = 5, λ₂ = 2.

For λ = 5: (A - 5I)v = [-1 2; 1 -2]v = 0 → v = (2, 1)

For λ = 2: (A - 2I)v = [2 2; 1 1]v = 0 → v = (1, -1)

(b) tr = 0, det = 1. λ = (0 ± √(-4))/2 = ±i. Complex eigenvalues λ = ±i.

This is rotation by 90° — no real eigenvectors.

(c) tr = 6, det = 9. λ = (6 ± √(36-36))/2 = 3. λ = 3 (double root).

For λ = 3: (A - 3I)v = [0 1; 0 0]v = 0 → v = (1, 0) only.

Only one eigenvector — not diagonalizable.

Exercise 2: Verification

For $A = \begin{pmatrix} 1 & 2 \\ 2 & 1 \end{pmatrix}$:

(a) Find eigenvalues using tr and det.

(b) Find eigenvectors.

(d) Construct P and verify P⁻¹AP = D.

Show Solution

(a) tr = 2, det = 1 - 4 = -3. λ = (2 ± √(4+12))/2 = (2 ± 4)/2. λ₁ = 3, λ₂ = -1.

(b) For λ = 3: [-2 2; 2 -2]v = 0 → v₁ = (1, 1)

For λ = -1: [2 2; 2 2]v = 0 → v₂ = (1, -1)

(c) v₁ · v₂ = 1 - 1 = 0 ✓ Orthogonal because A is symmetric.

(d) P = [1 1; 1 -1], P⁻¹ = [1/2 1/2; 1/2 -1/2]

P⁻¹AP = [3 0; 0 -1] = D ✓

Exercise 3: Properties

For a matrix with eigenvalues 2, 3, 5:

(a) What is the trace?

(b) What is the determinant?

(d) What are the eigenvalues of A⁻¹?

Show Solution

(a) tr = 2 + 3 + 5 = 10

(b) det = 2 × 3 × 5 = 30

(d) Eigenvalues of A⁻¹: 1/2, 1/3, 1/5

Exercise 4: Complex Eigenvalues

Find the eigenvalues of the rotation matrix:

$$R(60°) = \begin{pmatrix} 1/2 & -\sqrt{3}/2 \\ \sqrt{3}/2 & 1/2 \end{pmatrix}$$

Express them in the form e^(iθ). What does this tell you geometrically?

Show Solution

tr = 1, det = 1/4 + 3/4 = 1.

λ = (1 ± √(1-4))/2 = (1 ± √(-3))/2 = 1/2 ± i√3/2

In polar form: λ = cos(60°) ± i sin(60°) = e^(±iπ/3)

Geometrically: rotation by 60° has eigenvalues on unit circle at angles ±60°.

Exercise 5: Hückel Theory

For the allyl system (3 carbons in a row), the Hückel matrix is:

$$\mathbf{H} = \begin{pmatrix} \alpha & \beta & 0 \\ \beta & \alpha & \beta \\ 0 & \beta & \alpha \end{pmatrix}$$

(a) Setting α = 0 and β = -1, find the eigenvalues.

(b) Find the eigenvectors (MO coefficients).

Show Solution

(a) With α = 0, β = -1: H = [0 -1 0; -1 0 -1; 0 -1 0]

Characteristic polynomial: -λ³ + 2λ = -λ(λ² - 2) = 0

Eigenvalues: λ = 0, √2, -√2

(Energies: E = α + √2|β|, α, α - √2|β|)

(b) For λ = √2: c ∝ (1, √2, 1) (bonding)

For λ = 0: c ∝ (1, 0, -1) (nonbonding — node at center)

For λ = -√2: c ∝ (1, -√2, 1) (antibonding)

(c) Bonding: E = α + √2|β| (lowest energy).

Nonbonding: E = α.

Antibonding: E = α - √2|β| (highest energy).

Lecture 11: Symmetry

The eigenvectors of symmetric matrices are orthogonal. Molecular symmetry operations form groups. The mathematics of invariance.

Chemical Thinking: The Grammar of Reality

Lecture 10 — What Survives?

Lecture 10: What Survives?

The Hook

Eigenvector Visualizer

Recognition: The SAMENESS Primitive

When to Use Eigenvalues/Eigenvectors

Definition

Eigenvector and Eigenvalue

The Eigenspace

Finding Eigenvalues

The Characteristic Equation

The Characteristic Polynomial

2×2 Case

Special Cases

Finding Eigenvectors

Step-by-Step Eigenvalue Calculator

Properties of Eigenvalues

Algebraic and Geometric Multiplicity

Key Properties

Special Matrix Types

Complex Eigenvalues

When They Occur

Complex Eigenvalues: Rotation

Interpretation

Diagonalization

The Goal

The Method

When Diagonalization Fails

Why Diagonalization Matters

Chemistry Connection: Molecular Orbitals

The Secular Equation

Hückel Molecular Orbital Calculator

Chemistry Connection: Normal Modes

Vibrational Analysis

Chemistry Connection: Quantum Mechanics

The Eigenvalue Problem

Summary: Eigenvalue Toolkit

Quick Reference

Exercises

Exercise 1: 2×2 Eigenvalues

Exercise 2: Verification

Exercise 3: Properties

Exercise 4: Complex Eigenvalues

Exercise 5: Hückel Theory

Next