Definition

1. Noun:
   • (Mathematics) A scalar associated with a linear transformation: An eigenvalue of a square matrix is a special number (scalar) for which there exists a non-zero vector (the eigenvector) such that when the matrix multiplies that vector, the result is simply the scalar multiple of the original vector.
   • A root of the characteristic polynomial: Formally, it is any number λ such that for a given square matrix A, the matrix (A – λI) has a determinant of zero, where I is the identity matrix of the same dimension.

Usage Examples

• Noun:
  • Finding the eigenvalue of a square matrix is a fundamental step in diagonalization.
  • The stability of the system is determined by the eigenvalue of a square matrix derived from its differential equations.
  • If an eigenvalue of a square matrix is zero, the matrix is singular.

Advanced Usage

• "Eigenvalue problem": The mathematical problem of finding the eigenvalues and eigenvectors of a matrix.
  • Solving the eigenvalue problem is crucial for understanding the matrix's properties.
• "Dominant eigenvalue": The eigenvalue with the largest absolute value.
  • The power method is used to approximate the dominant eigenvalue of a large matrix.

Variants and Related Words

• Eigenvalue (n): The standard, shortened form of . Often used interchangeably.
  • The eigenvalues of the operator were complex numbers.
• Characteristic value (n): A synonym for eigenvalue.
• Latent root (n): Another synonym, less common.
• Eigenvector (n): The non-zero vector associated with a given eigenvalue.
  • Each eigenvalue has at least one corresponding eigenvector.

Synonyms

• Characteristic root
• Proper value (less common)

Related Phrases

• "Compute the eigenvalues of": To calculate the eigenvalues for a given matrix.
  • We need to compute the eigenvalues of the covariance matrix.
• "Eigenvalue decomposition": A factorization of a matrix into a canonical form.
  • Eigenvalue decomposition reveals the matrix's inherent structure.

Noun

1. (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant