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Understanding Positive Definite Matrices, Similar Matrices, and Jordan Form in Linear Algebra

Explore the concepts of positive definite matrices, similarity of matrices, and Jordan form in Linear Algebra. Learn about the properties of positive definite matrices, matrix addition and similarity, and the relationship between similar matrices. Understand the Jordan form and its importance in analyzing matrices with unique properties.

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Understanding Positive Definite Matrices, Similar Matrices, and Jordan Form in Linear Algebra

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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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