1. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN1 Lecture 28 is positive definite Similar matrices Jordan form

2. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN2 Positive definite means Except for Is the inverse of the symmetric positive matrix also the positive matrix? If A and B are positive definite, how about (A+B)? Now A is m by n matrix and rank(A)=n Is square and symmetric Is it positive definite? A and B are similar means For some matrix M For n by n matrices

3. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN3 Example : A is similar to B A is similar to B and Similar matrices have same eigenvalues has same eigenvalues as A In the same family

4. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN4 Eigenvector of matrix B is Bad case One small family has Big family includes Jordan form Can change to any value Best working matrix in this family Proof: Only one eigenvector Eigenvector of A

5. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN5 More members of family Every matrix has : 2 independent eigenvectors and 2 missing

6. NUU Department of Electrical Engineering Linear Algebra---Meiling CHEN6 2 independent eigenvectors and 2 missing Jordan block and are not equal Every square matrix A is similar to a Jordan matrix J # of blocks = # of eigenvectors Good case :