1、4 THE EIGENVALUE PROBLEM,Overview,In section 4.4 we move on to the general case, the eigenvalue problem for (nn) matrices. The general case requires several results from determinant theory, and these are summarized in section 4.2.,The eigenvalue problem is of great practical importance in mathematic

2、s and applications.,In section 4.1 we introduce the eigenvalue problem for the special case of (22) matrices; this special case can be handled using ideas developed in Chapter 1.,Core sections,The eigenvalue problem for (22) matrices Eigenvalues and the characteristic polynomial Eigenvectors and eig

3、enspaces Similarity transformations and diagonalization,4.1 The eigenvalue problem for (22) matrices,All scalars,Nonzero solution/ Infinitely many solution,1. The eigenvalue problem,The Geometric interpretation of Eigenvalue and eigenvector,The calculation of Eigenvalue and eigenvector,Homogeneous S

4、ystems,Eigenvalue and eigenvectors for (22) matrices,Example: Find all eigenvalues and eigenvectors of A, where,4.2 Determinants and the eigenvalue problem (omit),4.3 Elementary operations and determinants (omit),4.4 Eigenvalues and the characteristic polynomial,Example: Use the singularity test to

5、determine the eigenvalues of the matrix A, where,In this section we focus on part 1, finding the eigenvalues.,The characteristic polynomial,characteristic polynomial,characteristic equation,(1) an (nn) matrix can have no more than n distinct eigenvalues.,(2) an (nn) matrix always has at least one ei

6、genvalue.,Special Results,4.5 Eigenvectors and Eigenspaces,Eigenspaces and Geometric Multiplicity,Example Determine the algebraic and geometric multiplicities for the eigenvalues of A,Proof:,Corollary: Let A be an (nn) matrix. If A has n distinct eigenvalues, then A has a set of n linearly independe

7、nt eigenvectors.,4.7 Similarity Transformations And Diagonalization,In Chapter 1, we saw that two linear systems of equations have the same solution if their augmented matrices are row equivalent. In this chapter, we are interested in identifying classes of matrices that have the same eigenvalues.,D

8、efinition: The (nn) matrices A and B are said to be similar (denoted AB) if there is a nonsingular (nn) matrix S such that B=S-1AS.,Similarity,Theorem: If A and B are similar (nn) matrices, then A and B have the same eigenvalues. Moreover, these eigenvalues have the same algebraic multiplicity.,Note

9、: not generally have the same eigenvectors.,D is a diagonal matrix.,Diagonalization,Theorem: An (nn) matrix A is diagonalizable if and only if A possesses a set of n linearly independent eigenvectors.,Theorem: Let A be an (nn) matrix with n distinct eigenvalues. Then A is diagonalizable.,Whenever an

10、 (nn) matrix A is similar to a diagonal matrix, we say that A is diagonalizable.,Proof:,Proof:,Example Show that A is diagonalizable ,where,Orthogonal Matrices,A remarkable and useful fact about symmetric matrices is that they are always diagonalizable. Moreover, the diagonalization of a symmetric m

11、atrix A can be accomplished with a special type of matrix know as an orthogonal matrix.,Definition: A real (nn) matrix Q is called an orthogonal matrix if Q is invertible and Q-1=QT.,Theorem: Let Q be an (nn) orthogonal matrix. If X is in Rn, then |Q X |=| X |. If X and Y are in Rn , then (Q X)T(QY)

12、= X TY. det(Q)=1.,Diagonalizaiton of Symmetric Matrices,We conclude this section by showing that every symmetric matrix can be diagonalized by an orthogonal matrix.,Theorem: Let A be an (nn) real symmetric matrix, then the eigenvalues of A are real. (P319),Corollary: Let A be a real (nn) symmetric matri