44 Multiplicity of Eigenvalues

44 Multiplicity of Eigenvalues - Multiplicity of...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Multiplicity of Eigenvalues Learning Goals: to see the difference between algebraic and geometric multiplicity. We have seen an example of a matrix that does not have a basis’ worth of eigenvectors. For example. 1 1 0 1 ! " # $ % (note: this is not the Fibonacci matrix!). The characteristic polynomial of this matrix is (1 - λ ) 2 , so 1 is a double root of this polynomial. Finding the nullspace of A - λ I leads us to consider 0 1 0 0 ! " # $ % , whose nullspace only contains (1, 0) (and multiples of it). There is no place else to look for eigenvectors, so this matrix does not have an adequate supply! This leads us to two definitions: Definition: the algebraic multiplicity of an eigenvalue e is the power to which ( λ e ) divides the characteristic polynomial. Definition: the geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors associated with it. That is, it is the dimension of the nullspace of A eI . In the example above, 1 has algebraic multiplicity two and geometric multiplicity 1.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/22/2011 for the course ME 3 taught by Professor Prof.ramachandran during the Spring '11 term at Indian Institute of Technology, Kharagpur.

Page1 / 2

44 Multiplicity of Eigenvalues - Multiplicity of...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online