What is it?

Eigenvalues are a fundamental concept in linear algebra.


Given a square matrix A, an eigenvalue is a scalar Ξ» that satisfies the equation:

Aβˆ—v=Ξ»βˆ—vA*v=Ξ» * v


  • v is a non-zero vector called the eigenvector associated with the eigenvalue Ξ». In other words, when a matrix A operates on its eigenvector, the result is a scaled version of the eigenvector itself.

The important properties of eigenvalues and eigenvectors

  1. Multiplicity: An eigenvalue may have a multiplicity greater than 1, which indicates that there are multiple linearly independent eigenvectors associated with that eigenvalue.

  2. Eigenvalue equation: The equation A * v = Ξ» * v can be rewritten as (A - Ξ»I) * v = 0, where I is the identity matrix. This equation implies that the matrix (A - Ξ»I) is singular, i.e., it has a determinant of zero.

  3. Determinant and trace: The sum of the eigenvalues of a matrix A is equal to its trace (the sum of the diagonal elements) and the product of its eigenvalues is equal to its determinant.

  4. Diagonalizability: A matrix A is said to be diagonalizable if it has a full set of linearly independent eigenvectors. In this case, A can be expressed as A = P * D * P^(-1), where P is a matrix consisting of the eigenvectors of A, and D is a diagonal matrix with the corresponding eigenvalues on its diagonal.

Eigenvalues have numerous applications in various fields

  • In quantum mechanics, eigenvalues and eigenvectors play a fundamental role in determining the possible energy levels and corresponding states of a quantum system.

  • Eigenvalue decomposition is used in image processing for tasks like image compression and denoising.

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