When the geometric multiplicity is less than the

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: tive. The column space or range of a matrix is defined to be the span of its column vectors. If then The rank of B is the dimension of the range: Then the geometric multiplicity of is . The null space or kernel of a matrix B is the set of vectors that the matrix send to zero: . For example, consider an eigenvalue and eigenvector of matrix . These satisfy or equivalently . Then . The dimension of the kernel is called the nullity of a matrix: . 8 If matrix has columns, a theorem of linear algebra implies that . Continue the example. If the geometric multiplicity of is eigenvectors associated with and . then has linearly independent Return to the analysis of the boundary states in the -plane on the parabola where is the trace of and is the determinant of . Then [6] implies that the matrix an eigenvalue of algebraic multiplicity 2. , has When the geometric multiplicity of is 2, . That is, for every vector . This implies . Every nonzero vector is an eigenvector and the solution of is uniform expansion away from the origin, for , or uniform contraction toward the origin, for . When the geometric multiplicity of is 1, the matrix has only a single eigenvector. This means [4] cannot be the general solution of the initial value problem. The solution will be obtained in section 2.6. 2.3 Exponentials of Operators Let be a vector space. We may have or , but results in this section also apply to infinite dimensional vector spaces. An operator maps a vector to some other vector . A linear operator T satisfies linear superposition and scaling. If and and are scalars then . If and is a basis for then T will be represented by a matrix in that basis. In this case may replaced by in the equations below. We write , the space of linear operators on is itself an dimensional vector space. Notice that a matrix representing has components. If then operator norm represents the Euclidean norm: . We will also define an The operator norm satisfies the usual properties for norms. In particular, the triangle inequality is satisfied. If S and T are operators on then . [The matrix norm is a continuous function of its argument. See Hirsch and Smale p. 78.] 9 A linear operator also satisfies since, for . For any integer , we have by the same logic. , Since this is true for all [6] we conclude . An operator for which is said to be bounded. Example. Suppose . On a finite dimensional vector space the sphere is comp...
View Full Document

This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .

Ask a homework question - tutors are online