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If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication =, where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix—for example by diagonalizing it.
Let A be a square n × n matrix with n linearly independent eigenvectors q i (where i = 1, ..., n).Then A can be factored as = where Q is the square n × n matrix whose i th column is the eigenvector q i of A, and Λ is the diagonal matrix whose diagonal elements are the corresponding eigenvalues, Λ ii = λ i.
Given an n × n square matrix A of real or complex numbers, an eigenvalue λ and its associated generalized eigenvector v are a pair obeying the relation [1] =,where v is a nonzero n × 1 column vector, I is the n × n identity matrix, k is a positive integer, and both λ and v are allowed to be complex even when A is real.l When k = 1, the vector is called simply an eigenvector, and the pair ...
The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
Then an eigenvector and the corresponding eigenvalue must satisfy the equation =, or, equivalently (since =), = where is the identity matrix, and (although the zero vector satisfies this equation for every , it is not considered an eigenvector).
A substochastic matrix is a real square matrix whose row sums are all ; In the same vein, one may define a probability vector as a vector whose elements are nonnegative real numbers which sum to 1. Thus, each row of a right stochastic matrix (or column of a left stochastic matrix) is a probability vector.
Its eigenvalues are either 0 or 1: if is a non-zero eigenvector of some idempotent matrix and its associated eigenvalue, then = = = = =, which implies {,}. This further implies that the determinant of an idempotent matrix is always 0 or 1.
This is the matrix multiplication Ab = c written in summation notation and is a matrix equivalent of the operator D acting upon the function f(t) expressed in the orthonormal basis. If f(t) is an eigenfunction of D with eigenvalue λ, then Ab = λb.