# Eigenvalues and Eigenvectors

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/eigenvalues-and-eigenvectors

## Eigenvalues & Eigenvectors

Think of a transformation as a chaotic warp. Yet, some vectors only stretch—they never rotate. An **eigenvector** $\vec{v}$ of matrix $A$ is a nonzero vector satisfying $A\vec{v} = \lambda\vec{v}$.

The scalar $\lambda$ is the **eigenvalue**, scaling that vector. **Common pitfall**: Eigenvectors only keep their **line**. Eigenvalues can be negative (flips), zero (collapses), or fractions (shrinks).

| Property | Meaning |
| --- | --- |
| $\lambda > 0$ | Stretches or shrinks |
| $\lambda < 0$ | Flips direction |
| $\lambda = 0$ | Collapses to origin |

## Solving & Diagonalisation

Find eigenvalues via the **characteristic equation**: $\det(A - \lambda I) = 0$. For each root $\lambda_i$, find eigenvectors by solving $(A - \lambda_i I)\vec{v} = \vec{0}$.

**Diagonalisation**: If $A$ has $n$ independent eigenvectors, $A = PDP^{-1}$, where $D$ holds eigenvalues and $P$ holds eigenvectors as columns. This makes computing $A^k = PD^kP^{-1}$ trivial.

> **Example**: $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ yields $\lambda^2 - 7\lambda + 10 = 0$, giving eigenvalues $5$ and $2$.

## Practice questions

6 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. A transformation stretches the plane. Vector $v \ne 0$ satisfies $Av = 3v$. What is geometrically special about $v$’s **direction**?

A. $v$ is unaffected by the transformation entirely
B. The transformation keeps that direction fixed, stretching 3×
C. The transformation rotates $v$ by exactly 3 radians
D. $v$ is perpendicular to everything the matrix produces

**Answer:** B. The transformation keeps that direction fixed, stretching 3×

**Why:** Eigenvectors mark the invariant axes of a transformation, the directions it cannot turn, only rescale (by the eigenvalue). Finding them reveals a transformation’s skeleton: along those axes, matrix action reduces to simple multiplication.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/a-transformation-stretches-the-plane-vector-v-0-satisfies-av-3v

### 2. An eigenvector’s direction is unchanged by the transformation while every other direction turns. Why does that make eigenvectors useful for describing dynamics?

A. Because a matrix has as many eigenvectors as it has rows
B. Because eigenvectors are always perpendicular to one another
C. Because along an eigenvector the map is simple multiplication
D. Because they are the only directions in which the transformation is defined

**Answer:** C. Because along an eigenvector the map is simple multiplication

**Why:** Eigenvectors turn matrix powers into scalar powers. This is why normal modes, principal axes and Markov steady states are all eigenvector problems.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/an-eigenvectors-direction-is-unchanged-by-the-transformation-while

### 3. Order the steps to find eigenvalues and eigenvectors.

**Answer:**

1. Form A − $\lambda I$
2. Compute $\det(A - \lambda I) = 0$ (characteristic equation)
3. Solve for eigenvalues $\lambda$
4. For each $\lambda$, solve (A − $\lambda I$)$v = 0$ for eigenvectors

**Why:** First find the eigenvalues, then the corresponding eigenvectors.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/order-the-steps-to-find-eigenvalues-and-eigenvectors

### 4. A population redistributes each year by matrix $A$, and $v$ is an eigenvector with eigenvalue $\lambda = 2$. If the population starts at $v$ with total 100, what is the total after **3 years**? **Set the slider to your answer.**

**Answer:** 800 (within ±24)

**Why:** On an eigenvector, matrix dynamics collapse into scalar dynamics: $2^{3} = 8$-fold growth, so 800. This is why eigenvalues govern long-run behaviour, whichever eigenvalue is largest eventually dominates any starting mix.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/a-population-redistributes-each-year-by-matrix-a-and-v-is-an

### 5. A matrix acts on each vector below as noted. Sort: which are **eigenvectors**?

**Answer:**

- Eigenvector: $Av = 5v$ (stretched in place), $Av = -v$ (exactly reversed), $Av = 0$ (annihilated), $v \ne 0$
- Not an eigenvector: $Av = v + (1,0)$ (shifted sideways)

**Why:** Eigenvalue $\lambda$ can be any scalar: $-1$ flips, $0$ crushes (such $v$ spans the null space). The only disqualifier is leaving the original line, proportionality is the entire definition.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/a-matrix-acts-on-each-vector-below-as-noted-sort-which-are

### 6. Classify each matrix property in terms of its eigenvalues.

**Answer:**

- Related to eigenvalues: Trace = sum of eigenvalues, Determinant = product of eigenvalues, Invertibility (no zero eigenvalue)
- Not directly related: Number of rows, Matrix is symmetric

**Why:** Trace, determinant, and invertibility are all characterised by eigenvalues.

Page: https://tryals.app/practice/physics-i/eigenvalues-and-eigenvectors/classify-each-matrix-property-in-terms-of-its-eigenvalues