please solve Exercise 4.2,4.3,4.4 from the below
Exercise 4.1. Let AinC^(n imes n) satisfy A^(**)=-A. Show that the matrix I-A is invertible.
Then show that the matrix (I-A)^(-1)(I+A) is unitary.
Exercise 4.2. Let A=(a_(ij)) be a complex n imes n matrix. Assume that (:Ax,x:)=0 for all
xinC^(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=sqrt(-1) ) in various combinations.
Conclude that A=0.
Exercise 4.3. Let A,B be complex n imes n matrices such that (:Ax,x:)=(:Bx,x:) for all
xinC^(n). Use the previous exercise to prove that A=B.
Exercise 4.4. Find a real 2 imes 2 matrix A!=0 such that (:Ax,x:)=0 for all xinR^(2). Thus
find two real 2 imes 2 matrices A and B such that (:Ax,x:)=(:Bx,x:) for all xinR^(2), but
A!=B.
Exercise 4.5. Find a real 2 imes 2 matrix A such that (:Ax,x:)>0 for all xinR^(2), but A is
not positive definite.
Exercise 4.1. Let A e Cnxn satisfy A* = -A. Show that the matrix I -- A is invertible. Then show that the matrix (I - A)-1(I + A) is unitary.
Exercise 4.2. Let A = (aii) be a complex n x n matrix. Assume that (Ax,x) = 0 for all x E Cn. Prove that (a) aii = 0 for 1 < i < n by substituting x = ei (b) aij = 0 for i # j by substituting x = pe;+qej then using (a) and putting p, q = 1,i (here i = /-1) in various combinations. Conclude that A = 0
Exercise 4.3. Let A,B be complex n n matrices such that (Ax,x) = (Bx,x) for all x e Cn. Use the previous exercise to prove that A = B.
Exercise 4.4. Find a real 2 x 2 matrix A 0 such that (Ax,x) = 0 for all x e R2. Thus find two real 2 x 2 matrices A and B such that (Ax,x) = (Bx,x) for all x e R2, but A B.
Exercise 4.5. Find a real 2 x 2 matrix A such that (Ax,x) > 0 for all x E R2, but A is not positive definite.
