Ñ3 0 1é
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 .
