Verify the special theorem for the hermitian matrix

Question

Questions:

Matrices

1. Show that for a real symmetric matrix A=A^T, its singular values are exactly the absolute values of its eigenvalues, i.e. for every eigenvalue λ, there is a singular value σ = |λ|

2. Find the singular decomposition of. A = [ 3  5]
                                                                  [ 4  0]

3. Verify the special theorem for the Hermitian matrix

                        A = [ 3 2i]
                              [-2i 6]

i.e. find scales λ¹ and λ² and orthonormal vectors u¹ and u² such that

                                A = λ₁u₁u₁^* + λ₂u₂u₂^*