The Eigenvalue-Eigenvector Method
of Finding Solutions


EXAMPLE: Let be a linear operator such that



Let also



Find and


EXAMPLE: Let be a linear operator such that



Find all nonzero vectors and all numbers such that




DEFINITION: An eigenvector of an matrix is a nonzero vector such that



for some scalar A scalar is called an eigenvalue of The set of all solutions of the equation



is called the eigenspace of corresponding to

DEFINITION: Let



then



is called the characteristic polynomial of and is called the characteristic equation of

EXAMPLE: Let



Then



is a characteristic polynomial,



is a characteristic equation; and are the eigenvalues of and



are the eigenvectors of where is any nonzero real number;



is the eigenspace of corresponding to



is the eigenspace of corresponding to .

EXAMPLE: Let



Find all eigenvalues, eigenvectors and bases for the corresponding eigenspaces.


EXAMPLE: Let



Find all eigenvalues, eigenvectors and bases for the corresponding eigenspaces.


EXAMPLE: Let



The eigenvalues are and . Find bases for the corresponding eigenspaces.


THEOREM 12: Any eigenvectors of with distinct eigenvalues respectively, are linearly independent.

We return now to the first-order linear homogeneous differential equation

                                   (1)


Our goal is to find linearly independent solutions . One can show that



is a solution of (1) if, and only if, is an eigenvalue and is an eigenvector of .  

EXAMPLE: Find the general solution of the system

                                                                   (2)



EXAMPLE: Find all solutions of the equation