# Eigenvalues and Eigenvectors

Business I · Mathematics II · https://tryals.app/learn/business-i/eigenvalues-and-eigenvectors-mathematics-ii

## The Directions That Don't Rotate

Eigenvalues answer the most fundamental question about a linear system: **does it grow, shrink, or stay the same, and in which directions?**

Given a square matrix $A$, an **eigenvector** $\mathbf{v}$ is a non-zero vector that $A$ merely scales rather than rotates:

$$A\mathbf{v} = \lambda \mathbf{v}$$

The scalar $\lambda$ is the **eigenvalue**, representing the stretch, compression, or flip factor.

To find them, rearrange as $(A - \lambda I)\mathbf{v} = \mathbf{0}$. For a non-zero solution, the matrix must be singular:

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

This is the **characteristic equation**, a polynomial whose roots are the eigenvalues.

## The $2 \times 2$ Shortcut

For any $2 \times 2$ matrix $A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$, skip full expansion using the trace ($a+d$) and determinant ($ad-bc$).

| Quantity | Formula | Purpose |
|---|---|---|
| Char. Equation | $\lambda^2 - tr(A)\lambda + \det A = 0$ | Direct setup |
| Sum | $\lambda_1 + \lambda_2 = tr(A)$ | Fast check |
| Product | $\lambda_1 \lambda_2 = \det A$ | Fast check |

**Economic and Data Applications:**
- **Input-output models:** Dominant eigenvalue shows if an economy yields a net surplus.
- **Dynamic systems:** **Stability** requires all $|\lambda| < 1$; otherwise, the system explodes.
- **Markov chains:** The stationary distribution matches $\lambda = 1$.
- **PCA:** Eigenvectors of the covariance matrix reveal directions of greatest variation.

## Practice questions

4 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. If all eigenvalues of a transition matrix have magnitude greater than 1, the dynamic system converges to a steady state.

**Answer:** False

**Why:** False. Convergence needs every eigenvalue's magnitude **below** 1: each application scales its eigen-direction by $\lambda$, so $|\lambda| > 1$ makes that component grow without limit and the system diverges. The largest magnitude decides stability, which is why it alone is worth computing.

Page: https://tryals.app/practice/business-i/eigenvalues-and-eigenvectors-mathematics-ii/if-all-eigenvalues-of-a-transition-matrix-have-magnitude-greater-than

### 2. Match each eigenvalue application to its field.

**Answer:**

- Leontief input-output → Dominant eigenvalue determines net surplus feasibility
- Dynamic stability → Eigenvalue magnitudes determine convergence or explosion
- PCA → Eigenvectors of covariance matrix are principal components
- Markov chains → Stationary distribution is the eigenvector for eigenvalue 1

**Why:** Eigenvalues answer the same question across economics, statistics, and dynamics: what are the system's natural directions and rates?

Page: https://tryals.app/practice/business-i/eigenvalues-and-eigenvectors-mathematics-ii/match-each-eigenvalue-application-to-its-field

### 3. An eigenvector is described as a direction the matrix cannot bend. What makes those directions worth finding?

A. They are the only directions in which the matrix is invertible
B. They minimise the numerical error in matrix computations
C. They are the directions in which the matrix has the largest effect
D. Along them the matrix acts as a single number

**Answer:** D. Along them the matrix acts as a single number

**Why:** Aᵏv = λᵏv, so a hundred steps of a transition process is one exponentiation rather than a hundred matrix products. Finding a coordinate system where a complicated operator becomes multiplication is the recurring move of linear algebra, and it is what diagonalisation formalises.

Page: https://tryals.app/practice/business-i/eigenvalues-and-eigenvectors-mathematics-ii/an-eigenvector-is-described-as-a-direction-the-matrix-cannot-bend

### 4. Which statements about eigenvalues are correct?

A. A 2x2 matrix always has exactly 2 distinct eigenvalues
B. Eigenvalues are roots of the characteristic polynomial
C. The product of eigenvalues equals the determinant
D. An eigenvector is scaled but not rotated by the matrix

**Answer:** B. Eigenvalues are roots of the characteristic polynomial; C. The product of eigenvalues equals the determinant; D. An eigenvector is scaled but not rotated by the matrix

**Why:** A 2x2 matrix can have a repeated eigenvalue (e.g., $\lambda = 3$ with multiplicity 2), so the third statement is false.

Page: https://tryals.app/practice/business-i/eigenvalues-and-eigenvectors-mathematics-ii/which-statements-about-eigenvalues-are-correct