Calculate eigenvalues and eigenvectors of 2x2, 3x3, 4x4, and 5x5 matrices
Eigenvalues and eigenvectors are fundamental concepts in linear algebra that reveal important properties of linear transformations.
| Matrix Type | Matrix | Eigenvalues |
|---|---|---|
| Rotation Matrix |
[0 -1] [1 0] |
0 + 1i, 0 - 1i |
| Skew-Symmetric |
[0 2] [-2 0] |
0 + 2i, 0 - 2i |
| Complex Matrix |
[1 i] [-i 1] |
0, 2 |
In linear algebra, an eigenvector of a square matrix A is a non-zero vector v such that A v = λ v, where λ (lambda) is a scalar known as the eigenvalue. Geometrically, eigenvectors represent directions that remain invariant under the linear transformation — they may be stretched, compressed, or reversed, but not rotated out of their span.
A · v = λ v ⇔ (A - λI) v = 0
The characteristic equation det(A - λI) = 0 yields eigenvalues. For a 2×2 matrix, the characteristic polynomial is λ² − (tr A)λ + det(A) = 0. For 3×3, it is -λ³ + tr(A) λ² - (sum of principal minors) λ + det(A)=0.
Eigenvectors are fundamental to quantum mechanics (states & observables), Google's PageRank algorithm, facial recognition (Eigenfaces), vibration analysis, and principal component analysis (PCA) in machine learning.
In data science, eigenvectors of the covariance matrix reveal principal components. The eigenvector associated with the largest eigenvalue indicates the direction of maximum variance. This calculator helps users comprehend how 2D point clouds stretch, which directly translates to understanding feature importance in machine learning pipelines. For 3x3, the same principle applies to three-dimensional data.
| Matrix type | Eigenvalue nature | Transformation behavior |
|---|---|---|
| Symmetric | Real, orthogonal eigenvectors | Stretch along orthogonal axes |
| Rotation (2D) | Complex conjugate | Pure rotation, no real eigenvectors |
| Defective (repeated eigenvalue) | May have fewer independent eigenvectors | Shear-like behavior |