## Applications to Voting and Social Choice

Linear algebra provides mathematical tools for analyzing voting systems, preference aggregation, and collective decision-making.

### Preference Vectors

In ranked voting, each voter expresses a preference ordering over candidates. These preferences can be represented as vectors or matrices:

- **Pairwise comparison matrix**: Entry a_ij counts how many voters prefer candidate i over candidate j
- **Score vectors**: Each candidate receives a numerical score based on ranking position

### Eigenvector-Based Ranking Methods

**Perron-Frobenius Theorem** and eigenvector analysis underlie several ranking methods:

**Kendall Method**: Uses the principal eigenvector of a pairwise comparison matrix to produce a collective ranking.

**PageRank Adaptation**: Similar to Google algorithm, eigenvector centrality ranks candidates based on the strength of preferences pointing to them.

### Voting as Linear Transformations

Different voting methods (plurality, Borda count, ranked pairs) can be viewed as linear transformations of preference data into social rankings.

### Borda Count Example

For n candidates, the Borda count assigns points (n-1, n-2, ..., 1, 0) to ranking positions. The total scores form a vector:

**s = B^T v**

where B encodes the ballot matrix and v is the vote distribution vector.

### Fair Division

Linear programming and matrix methods solve fair division problems: dividing resources among parties with different valuations to achieve equitable outcomes.