227A-HW2

Norms Distinct Eigenvalues, Orthogonal Eigenspaces Gram Schmidt Eigenvectors of a Symmetric Matrix Positive Semi-Definite Matrices SVD Transformation

26 flashcards · Mathematics · by John Viljoen

1. What is the definition of an Lp norm?

2. what is the definition of the L1 norm?

   sum of the absolute values

3. what is the definition of the infinity norm?

   the maximum absolute values

4. what is the cauchy schwartz inequality?

   for vectors u and v

5. what is the triangle inequality?

   for any Lp norm

6. Show that:

7. Show that:

8. Show that:

   square both sides, then:

9. Show that:

10. Show that:

11. what is the cardinality of a vector?

    the number of non-zero elements, also known as the L0 norm

12. show that for any non-zero x: where the L0 norm is the cardinality of x which is the number of non zero elements

13. what is an eigenpair

    an eigenvalue and its corresponding eigenvalue

14. therefore distinct eigenvalues correspond to orthogonal eigenvectors.

15. what is the algebraic multiplicity of an eigenvalue?

    The algebraic multiplicity of an eigenvalue λ of A is the number of times λ appears as a root of A's characteristic polynomial.

16. what is the geometric multiplicity of an eigenvalue?

    the geometric multiplicity of an eigenvalue lambda of A is the dimension of the eigenspace of A for lambda (=1 if lambda is non zero, =0 if lambda zero)

17. what do we know about the geometric and algebraic multiplicity of a symmetric matrix?

    they are equal

22. assume A = known:

23. Give 2 solutions.

26. Show that: