线性转换T的核空间.ppt

 Chapter 7 Eigenvalues and Eigenvectors 7.1 Eigenvalues and Eigenvectors 7.2 Diagonalization 7.3 Symmetric Matrices and Orthogonal Diagonalization 7.1 Eigenvalues and Eigenvectors Eigenvalue problem: Ex 1: (Verifying eigenvalues and eigenvectors) Thm 7.1: (The eigenspace of A corresponding to ?) Ex 3: (An example of eigenspaces in the plane) Find the eigenvalues and corresponding eigenspaces of For a vector on the y-axis Thm 7.2: (Finding eigenvalues and eigenvectors of a matrix A?Mn?n ) Ex 4: (Finding eigenvalues and eigenvectors) Ex 5: (Finding eigenvalues and eigenvectors) Find the eigenvalues and corresponding eigenvectors for the matrix A. What is the dimension of the eigenspace of each eigenvalue? Notes: Ex 6：Find the eigenvalues of the matrix A and find a basis for each of the corresponding eigenspaces. Thm 7.3: (Eigenvalues for triangular matrices) Eigenvalues and eigenvectors of linear transformations: Keywords in Section 7.1: eigenvalue problem: 特徵值問題 eigenvalue: 特徵值 eigenvector: 特徵向量 characteristic polynomial: 特徵多項式 characteristic equation: 特徵方程式 eigenspace: 特徵空間 multiplicity: 重數 The eigenspace of A corresponding to : Thus, the dimension of its eigenspace is 2. 15- * (1) If an eigenvalue ?1 occurs as a multiple root (k times) for the characteristic polynominal, then ?1 has multiplicity k. (2) The multiplicity of an eigenvalue is greater than or equal to the dimension of its eigenspace. 15- * Sol: Characteristic equation: Eigenvalue: 15- * is a basis for the eigenspace of A corresponding to 15- * is a basis for the eigenspace of A corresponding to 15- * is a basis for the eigenspace of A corresponding to 15- * If A is an n?n triangular matrix, then its eigenvalues are the entries on its main diagonal. Ex 7: (Finding eigenvalues for diagonal and triangular matrices) Sol: 15- * 15- * Ex 8: (Finding eigenvalues and eigenspaces) Sol: 15- * Notes: 15- * 15- * 15- * Lecture 15: Linear Transformation Today Matrices