eigenvalue of a square matrix

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eigenvalue of a square matrix

A student calculates the eigenvalue of a square matrix on a chalkboard.

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.
eigenvalue of a square matrix

A student calculates the eigenvalue of a square matrix on a chalkboard.

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

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