Description
1. Let A =
Ñ
3 0 1
0 2 1
0 0 2é
, determine whether A is nondefective, i.e., whether
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.
4. Use SVD to solve Ax = b. Let A = UΣV
T with U =
Ç√
2
2 −
√
2
√
2
2
2
√
2
2
å
, Σ =
Å
2 0
0 1ã
, V =
Å
0 1
1 0ã
, b =
Ç
−
3
2
√
2
−
√
2
2
å
.
