Eigenvalue Calculator
Find eigenvalues and eigenvectors of a 2x2 matrix with step-by-step characteristic polynomial derivation and full solution.
2x2 Matrix
[1 2]
Results
Eigenvalue Examples for Common Matrix Types
| Matrix Type | Matrix | Eigenvalues | Interpretation |
|---|---|---|---|
| Identity | [1 0; 0 1] | lambda = 1, 1 | All vectors are eigenvectors |
| Diagonal | [3 0; 0 -2] | lambda = 3, -2 | Read from diagonal directly |
| Rotation 90° | [0 -1; 1 0] | lambda = i, -i | Complex - no real eigenvectors |
| Shear | [1 1; 0 1] | lambda = 1, 1 | Repeated eigenvalue |
| Symmetric | [4 2; 2 1] | lambda = 5, 0 | Real eigenvalues guaranteed |
Understanding Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are among the most powerful concepts in linear algebra. The word "eigen" comes from German and means "own" or "characteristic" - eigenvalues are the characteristic values that describe a matrix's fundamental behavior.
Think of a matrix as a transformation of space - it stretches, rotates, and shears vectors. Most vectors change both direction and magnitude when transformed. Eigenvectors are special: they only change magnitude (get scaled by the eigenvalue) but keep their direction. The eigenvalue tells you exactly how much scaling occurs - a positive eigenvalue greater than 1 stretches, between 0 and 1 compresses, negative flips direction.
Real-World Applications
Google's original PageRank algorithm finds the eigenvector of the web's link matrix - the eigenvector corresponding to the largest eigenvalue gives each page's rank. Principal Component Analysis (PCA) in data science finds the eigenvectors of a covariance matrix to identify the directions of greatest variance. Quantum mechanics uses eigenvalues to describe the measurable states of physical observables.