eigenvalue of a matrix
Noun: 1. (Mathematics): A scalar (a number) associated with a given square matrix. Specifically, it is a number λ (lambda) for which there exists a non-zero vector v such that when the matrix multiplies that vector, the result is simply the scalar λ times the original vector v. This relationship is the defining equation: A v = λ v. Equivalently, λ is an eigenvalue if the matrix (A - λI), where I is the identity matrix, has a determinant of zero.
The term is used in linear algebra to describe the special scalars that reveal fundamental properties of a linear transformation represented by the matrix, such as its stretching factors and directions of invariance. * Finding the eigenvalue of a matrix is a central problem in linear algebra. * The stability of a system is often determined by the eigenvalues of its coefficient matrix.
- In the equation A x = λ x, the scalar λ is the eigenvalue of the matrix A.
- To find the eigenvalues of a matrix, one must solve the characteristic equation det(A - λI) = 0.
- A symmetric matrix is guaranteed to have real eigenvalues.
- Multiplicity: An eigenvalue can have an (how many times it is a root of the characteristic polynomial) and a (the dimension of its associated eigenspace).
- Spectrum: The set of all eigenvalues of a matrix is called its .
- Dominant Eigenvalue: In many applied contexts, the eigenvalue with the largest absolute value (the or eigenvalue) is of particular importance, for example, in power iteration methods or analyzing population growth models.
- Eigenvalue (n): The more common, shortened form of the term. (e.g., "Calculate the eigenvalues.")
- Characteristic Value (n): A direct synonym for eigenvalue.
- Proper Value (n): Another, less common synonym.
- Latent Root (n): A synonym occasionally used in certain fields like statistics.
- Eigenvector (n): The non-zero vector v associated with a specific eigenvalue λ in the equation A v = λ v.
- Eigenspace (n): The set of all eigenvectors associated with a particular eigenvalue λ, along with the zero vector.
- Characteristic value
- Proper value
- Latent root
- Eigenvalue decomposition: A matrix factorization that expresses a matrix in terms of its eigenvalues and eigenvectors.
- Eigenvalue problem: The mathematical problem of finding the eigenvalues (and often eigenvectors) of a matrix.
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(mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant
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