Eigenvalue and Eigenvectors(特征值和

This chapter begins the "second half" of linear algebra. The first half was about . The new problem will stil be solved by simplifying a matrix --- making it diagonal if possible. The basic step is no longer to subtract a multiple of one row from another: Elimination changes the eigenvalues, which we don't want.

Eigenvalue equation:

The number λ is an eigenvalue of the matrix A
and the vector x is the associated eigenvector

1. Introduction

• is a nonlinear equation
• 
  The vector x is in nullspace of
  The number λ is chosen so that has a nullspace
• The number λ is an eigenvalue of A if and only if is singular:
  
  if is invertible, then , then x = 0; however, we need the nullspace of contain vectors other than zero. So if and only if is singular.
• Eigenspace: all vectors in the nullspace of
• Steps in solving :
  • Compute the determinant of . With λ subtracted along the diagonal, this determinant is a polynomial of degree n. It starts with
  • Find the roots of this polynomial. The n roots are the eigenvalue of A
  • For each eigenvalue solve the equation . Since the determinant is zero, there are solutions other than x=0. These are the eigenvectors.
• A zero eigenvalue signals that A is singular(not invertible); its determinant is zero. Invertible matrices have all
• The eigenvalue are on the main diagonal when A is triangular; so we need to transform A into a diagonal or triangular matrix without changing its eigenvalues. However, Gaussian factorizstion A=LU is not suited to this purpose. The eigenvalue of U may be visible on the diagonal, but they are not the eigenvalues of A.
• The sum of the n eigenvalues equals the sum of the n diagonal entries:
  Trace of
  The Product of the n eigenvalues equals the determinant of A.