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What is the Eigenvalue in a Matrix?

Published in Linear Algebra 2 mins read

An eigenvalue of a matrix is a scalar value that represents how much an eigenvector is scaled when transformed by that matrix.

Here's a breakdown:

  • Definition: If A is a matrix, v is a non-zero vector (eigenvector), and λ is a scalar (eigenvalue), then the equation Av = λv holds true. This means that when the matrix A is multiplied by the eigenvector v, the resulting vector is simply a scaled version of the original eigenvector v, where the scaling factor is the eigenvalue λ.

  • Mathematical Representation: The eigenvalue equation is expressed as:
    Av = λv

    where:

    • A is the matrix.
    • v is the eigenvector.
    • λ is the eigenvalue.

  • Geometric Interpretation: Eigenvalues represent the factors by which corresponding eigenvectors are stretched or shrunk by a linear transformation. If an eigenvalue is:

    • Positive: The eigenvector is stretched in the same direction.
    • Negative: The eigenvector is stretched in the opposite direction.
    • Zero: The eigenvector is mapped to the zero vector.
    • Greater than 1: The eigenvector is stretched.
    • Between 0 and 1: The eigenvector is shrunk.

  • How to Find Eigenvalues:

    1. Start with the equation Av = λv.
    2. Rewrite it as (A - λI)v = 0, where I is the identity matrix of the same size as A.
    3. For a non-trivial solution (v ≠ 0), the determinant of (A - λI) must be zero: det(A - λI) = 0.
    4. Solve the characteristic equation det(A - λI) = 0 for λ. The solutions for λ are the eigenvalues of the matrix A.

  • Example:
    Let's say we have matrix A = [[2, 1], [1, 2]].
    To find the eigenvalues:

    1. A - λI = [[2-λ, 1], [1, 2-λ]]
    2. det(A - λI) = (2-λ)(2-λ) - (1)(1) = λ² - 4λ + 3 = 0
    3. Solving for λ: (λ - 3)(λ - 1) = 0. So, λ = 3 and λ = 1.
      Therefore, the eigenvalues of matrix A are 3 and 1.

  • Significance: Eigenvalues and eigenvectors are crucial in various applications, including:

    • Principal Component Analysis (PCA): Used for dimensionality reduction.
    • Vibrational Analysis: Determining natural frequencies of structures.
    • Quantum Mechanics: Describing energy levels of particles.
    • Stability Analysis of Systems: Determining the stability of dynamic systems.

In essence, an eigenvalue tells you how much an eigenvector changes in magnitude (and potentially direction, if negative) when a matrix transformation is applied.

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