MATH307 Group Homework8 Solved

Description

Instructions: Read textbook pages 149 to 155 before working on the homework problems. Show all steps to get full credits.

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1. Let A = 0 2 1 , determine whether A is nondefective, i.e., whether

0 0 2

the algebra multiplicity and the geometric multiplicity are identical for all eigenvalues.

2. Let A be a Hermitian matrix, prove that eigenvectors corresponding to distinct eigenvalues of A are orthogonal. Note this is stronger than eigenvectors of distinct eigenvalues of a general matrix are linearly independent.
3. Let A = UΣV ^∗ be a singular value decomposition of A, prove that V (Σ^∗Σ)V ^∗ is an eigendecomposition of A^∗A.

√         √

Ç 2 − 2å

4. Use SVD to solve Ax = b. Let A = UΣV T with

Å2 0ã    Å0 1ã Çå ,V         ,b         .