# Eigenvalues & Eigenvectors

Canonical: https://manyspace.aporia.systems/topics/eigenvalues-eigenvectors-mth-linalg-162
Updated: 2026-10-13

Mathematics · Linear Algebra · MTH-LINALG-162

- Difficulty: 3 / 5
- Importance: 5 / 5
- Study time: 3–6 h

characteristic polynomial, eigenspaces, diagonalization

## Why it matters

Central to dynamical systems, quantum mechanics, PCA, PageRank and stability analysis.

## Where it is used

Stability of mechanical systems, PCA in data science, energy levels in QM, Google's PageRank.

## How you know you have learned it

Complete 3 independent examples correctly + explain common failure modes.

## What you need first

- [Determinants](https://manyspace.aporia.systems/topics/determinants-mth-linalg-157) (required)
- [Linear Maps & Matrices](https://manyspace.aporia.systems/topics/linear-maps-matrices-mth-linalg-161) (required)
- [Polynomial Functions](https://manyspace.aporia.systems/topics/polynomial-functions-mth-precalc-077) (helpful)
- [Direct Proof](https://manyspace.aporia.systems/topics/direct-proof-mth-proofs-104) (helpful)

## What it opens

- [Similarity & Jordan Form Intro](https://manyspace.aporia.systems/topics/similarity-jordan-form-intro-mth-linalg-163)
- [Spectral Theorem](https://manyspace.aporia.systems/topics/spectral-theorem-mth-linalg-166)
- [Systems of ODEs](https://manyspace.aporia.systems/topics/systems-of-odes-mth-ode-191)
- [Schrodinger Equation](https://manyspace.aporia.systems/topics/schrodinger-equation-phy-qm-107)
- [Angular Momentum](https://manyspace.aporia.systems/topics/angular-momentum-phy-qm-115)
- [The Multivariate Gaussian](https://manyspace.aporia.systems/topics/the-multivariate-gaussian-aiml-prb-008)
- [Principal Component Analysis](https://manyspace.aporia.systems/topics/principal-component-analysis-aiml-uns-005)

## Resources

- Linear Algebra Done Right — Sheldon Axler
- Linear Algebra Done Right — Sheldon Axler
- Introduction to Linear Algebra — Gilbert Strang