1 3 points Let V be an ndimensional real space and let T be

1. (3 points) Let V be an n-dimensional (real) space and let T be a linear trans vector formation from V into V such that the range and null space of T are identical. Prove that n, the dimension of V is an even integer. 2. (4 points) Prove that the eigenvalues of a Hermitian matrix H must be real numbers

Solution

2)

Let H be a Hermitian matrix.

Then, by definition:

H = H   where A denotes the conjugate transpose of H.

Let eigenvalue 0 such as

      Hv =v

(Hv)* =(v)*

(v*H*)=(*v*)

Right-multiply both sides by  v,

(v*H*v)=(*v*v )

     But H*=H

(v*Hv )=(*v*v )

(v*v)=(*v*v)

(v*v )=(*v*v )

= *

R

Therefore,

eigen values of Hermitian matrix H are always real numbers.