If the application of an operator to a function gives back the original function multiplied by some constant, then this constant is known as the eigenvalue of that function. In quantum mechanics, eigenvalues often correspond to observables.
and eigenvector. If a (scalar) value, t, satisfies Ax = tx for some vector, x not= 0, it is an eigenvalue of the matrix A, and x is an eigenvector. In mathematical programming this arises in the context of convergence analysis, where A is the hessian of some merit function, such as the objective or Lagrangian.
a mathematical expression associated with each component and represents the amount of variance explained by that component
a unique scalar which when it multiplies an eigenvector, produces a resultant vector equivalent to the operator of the eigen-equation applied to the same eigenvector
A number is an eigenvalue of the square matrix if there exists a nonzero vector such that . Eigenvalues are used to determine the stability of critical points of systems of first-order, autonomous differential equations.
The variance associated with a particular eigenvector (or principal component).
the variance in a set of variables explained by a factor or component, and denoted by lambda . An eigenvalue is the sum of squared values in the column of a factor matrix, or where a ik is the factor loading for variable i on factor k, and m is the number of variables. In matrix algebra the principal eigenvalues of a correlation matrix are the roots of the characteristic equation
and Eigenvector (see PCA, EOF) of a square matrix B are a scalar and a vector that satisfies the equation Be = , where is the eigenvalue and is the eigenvector. [pg 369, 3
The change in length that occurs when the corresponding eigenvector is multiplied by its matrix.
scalar value λ that permits nonzero solutions of equations of the form where is a linear operator and where can represent a vector or a function that is subject to certain boundary conditions. When is a vector, represents a matrix and is termed an eigenvector. When is a function, can represent a differential or integral form, in which case is called an eigenfunction.
