Numerical linear algebra-complex nxn matrix

Question

Problems:

Numerical Linear Algebra : Complex nxn Matrix

1. Let A = (a_ij ) be a complex n × n matrix. Assume that {Ax, x}= 0 for all x ∈ ?ⁿ.
Prove that

(a) a_ii = 0 for 1 ≤ i ≤ n by substituting x = e_i

(b) a_ij = 0 for i ≠ j by substituting x = pe_i+qe_j then using (a) and putting p, q = ±1, ±i
(here i =√-1) in various combinations

Conclude that A = 0.

2. Find a real n × n matrix A ≠ 0 such that {Ax, x}>0 for all x ∈ ℜⁿ.

3. Find a real n×n matrix A such that {Ax, x}> 0 for all x ≠ 0, but A is not symmetric.

Hence, the symmetry requirement in Definition 12.9 cannot be dropped in the real case.

4. Let A ∈ ℜⁿ be given, symmetric and positive definite. Define A₀ = A, and consider the sequence of matrices defined by

                             A_k = G_kG^t_k   and   A_k+1 = G^t_kG_k
where A_k = G_kG^t_k is the Cholesky factorization for A_k. Prove that the Ak all have the same eigenvalues.

5. Let A ∈ ?nxn and J a Jordan canonical form of A. Show that A has a square root (in the complex sense!) if and only if so does J. Show that if J is diagonal, then both J and A have square roots.

[Extra credit] Let J = ( λ  1) be a nondiagonal Jordan block. Show that J has a square root if and only if λ ≠ 0.
                                 ( 0  λ)