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Lecture 28

Lecture 28. is positive definite Similar matrices Jordan form. 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.

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Lecture 28

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  1. Lecture 28 • is positive definite • Similar matrices • Jordan form Linear Algebra---Meiling CHEN

  2. 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 it positive definite? Is square and symmetric For n by n matrices A and B are similar means For some matrix M Linear Algebra---Meiling CHEN

  3. Example : A is similar to A is similar to B and has same eigenvalues as A B Similar matrices have same eigenvalues In the same family Linear Algebra---Meiling CHEN

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

  5. More members of family Every matrix has : 2 independent eigenvectors and 2 missing Linear Algebra---Meiling CHEN

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

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