eigenvalue
- Noun:
- A special number associated with a linear transformation: In mathematics, an eigenvalue is a special scalar (number) associated with a given square matrix or linear transformation. For a matrix, it is a number such that when it is subtracted from each diagonal element (via a specific calculation), the resulting matrix has a determinant of zero. This indicates the transformation has a non-trivial direction (eigenvector) that is only scaled, not rotated.
- Noun:
- The dominant eigenvalue of the transition matrix determines the system's long-term behavior.
- To solve the differential equation, we first need to calculate the eigenvalues of the coefficient matrix.
- An eigenvalue of zero means the transformation compresses space along that eigenvector's direction.
"To find/compute the eigenvalues of a matrix": This is a standard phrase in linear algebra for the process of solving the characteristic equation.
- The first step in diagonalizing a matrix is to find its eigenvalues.
"Eigenvalue decomposition": Refers to the factorization of a matrix into a canonical form, revealing its eigenvalues and eigenvectors.
- Principal Component Analysis relies on eigenvalue decomposition of the covariance matrix.
"Eigenvalue problem": A mathematical problem where one must find the eigenvalues (and often eigenvectors) of a given operator or matrix.
- Solving the quantum harmonic oscillator involves a classic Sturm-Liouville eigenvalue problem.
Eigenvector (n): The non-zero vector associated with a given eigenvalue that only gets scaled when the linear transformation is applied.
- For each eigenvalue, there is a corresponding eigenvector.
Eigenspace (n): The set of all eigenvectors associated with a particular eigenvalue, along with the zero vector.
- The dimension of the eigenspace is the geometric multiplicity of the eigenvalue.
Spectrum (n): In this context, the set of all eigenvalues of a matrix or operator.
- The stability of the system depends on the spectrum of the Jacobian matrix.
- Characteristic value: A direct synonym, stemming from the "characteristic equation" used to find eigenvalues.
- Proper value: An older, less common term with the same meaning (used in some historical contexts).
- Latent root: A term sometimes used in factor analysis and statistics.
- "Eigenvalue equation": The defining equation Av = λv, where A is a matrix, λ is an eigenvalue, and v is its eigenvector.
- The eigenvalue equation captures the essence of the transformation's invariant directions.
"Diagonalizable matrix": A matrix is diagonalizable if it has a complete set of eigenvectors, which form a basis. The diagonal entries of the similar diagonal matrix are the eigenvalues.
- A symmetric matrix is always diagonalizable with real eigenvalues.
"Trace and determinant relation": For a 2x2 matrix, the sum of the diagonal elements (trace) equals the sum of the eigenvalues, and the determinant equals the product of the eigenvalues. This relationship generalizes.
- You can check your calculated eigenvalues by verifying their sum equals the trace.
- (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant