Eigenvalues and Eigenvectors Explained

What are Eigenvalues and Eigenvectors?

In linear algebra, when we apply a linear transformation (represented by a matrix A) to a vector x, the resulting vector Ax usually changes direction and magnitude.

However, for any given square matrix A, there often exist special non-zero vectors called eigenvectors. When the matrix A is multiplied by one of its eigenvectors v, the resulting vector Av is simply a scaled version of the original eigenvector v. The vector doesn't change its direction (it stays on the same line through the origin), it only gets stretched or shrunk (and possibly flips direction if the scaling factor is negative).

Mathematically, this relationship is expressed as: Av = λv

• A is the square matrix representing the linear transformation.
• v is the non-zero eigenvector.
• λ (lambda) is the corresponding scalar value called the eigenvalue. It represents the factor by which the eigenvector is scaled (stretched, shrunk, or flipped).

Significance

Eigenvalues and eigenvectors reveal fundamental properties about the linear transformation represented by the matrix A.

• They indicate the directions in which the transformation acts purely as a scaling (stretching/shrinking).
• The eigenvalues tell you the magnitude of that scaling in those directions.
• They are crucial in many areas of science and engineering, including:
  • Principal Component Analysis (PCA) in data analysis and dimensionality reduction.
  • Analyzing stability in dynamical systems and control theory.
  • Solving systems of differential equations.
  • Understanding vibrational modes in mechanics and quantum mechanics.

This simulation helps visualize how a 2x2 matrix transforms vectors and highlights its eigenvectors.

Interactive 2x2 Matrix Simulation