Eigenvalue and eigenvector pdf

Eigenvalues and Eigenvectors Let A be an n n square matrix. Then x 7!Ax maps Rn to Rn. Its simple part: images Ax that are parallel" to x. Def: When Ax = x has a non-zero vector solution x: is called an eigenvalue of A. x is called an eigenvector of A corresponding to . Notes: (i) eigenvector must be non-zero. Lecture 14 (Eigenvalue & Eigenvector) De nition 1. Let V be a vector space over F and T: V !V be a linear transformation. Then 1. a scalar 2F is said to be an eigenvalue or characteristic value of T if there exists a non-zero vector v2V such that Tv= v. 2. a non-zero vector vsatisfying Tv= vis called eigenvector or characteristic vector of T In particular, 0 is an eigenvalue for the eigenvector w w0. 8. Suppose that T: R2!R2 is a linear transformation. Further suppose that x;y 2R2 are linearly independent eigenvectors of Tbut they have the same eigenvalue . Show that every vector in R2 is an eigenvector of T (associated to the same eigenvalue) and also that the characteristic poly- If Ais an n nmatrix, a generalized eigenvector of A corresponding to the eigenvalue is a nonzero vector x satisfying (A I)p x = 0 for some positive integer p. Equivalently, it is a nonzero element of the nullspace of (A I)p. Example I Eigenvectors are generalized eigenvectors with p= 1. I In the previous example we saw that v = (1;0) and Eigenvalue and Eigenvector Homework Olena Bormashenko November 14, 2011 For each of the matrices Abelow, do the following: 1. Find the characteristic polynomial of A, and use it to nd all the eigen- values of A. 2. State the algebraic multiplicity of each eigenvalue. 3. One can therefore determine the line of reflection by computing the eigenvector that corresponds to λ = 1, cosθ sinθ sinθ −cosθ x y = x y . (13) If θ = 0 (mod 2π), then any vector of the form (x 0) is an eigenvector corresponding to the eigenvalue λ = 1. This implies that the line of reflection is the x-axis, which corresponds to • if v is an eigenvector of A with eigenvalue λ, then so is αv, for any α ∈ C, α 6= 0 • even when A is real, eigenvalue λ and eigenvector v can be complex • when A and λ are real, we can always find a real eigenvector v associated with λ: if Av = λv, with A ∈ Rn×n, λ ∈ R, and v ∈ Cn, then Aℜv = λℜv, Aℑv = λℑv which is the eigenvalue/eigenvector problem by definition. The eigenvalue/eigenvector pairs are orthogonal and the system evolves as t n n x ce tv1 ce nv 1 1 The coefficients c 1…c n are determined by taking inner products of both sides of the equation with eigenvectors vi at time t=0. For driven systems, convolution integrals are Eigenvalues and eigenvectors play an important part in the applications of linear algebra. The naive method of nding the eigenvalues of a matrix involves nding the roots of the characteristic polynomial of the matrix. In industrial sized matrices, however, this method is not feasible, and the eigenvalues must be obtained by other means. The Eigenvalue Problem The Basic problem: n×nn, x 6= 0 such that: Ax = λx. λ is an eigenvalue and x is an eigenvector of A. An eigenvalue and corresponding eigenvector, (λ,x) is called an eigenpair. The spectrum of A is the set of all eigenvalues of A. To make the definition of a eigenvector precise we will often normalize the vector so it is said to be an eigenvector if there is a scalar λ, such that Av = λv. (1) The scalar λ is referred to as an eigenvalue of A. The eigenvalue may be a complex number and the eigenvector may have complex entries. If