aline science forum beginner
Joined: 29 Nov 2006
Posts: 2

Posted: Wed Nov 29, 2006 2:37 am Post subject:
Linear operator and determinant



Hi, I am having trouble with the following exercise.
1Let V be the vector space of all nxn matrices over the field of
complex numbers, and let B be a fixed nxn matrix over C. Define a
linear operator M_B on V by M_B(A) = BAB*, where B* is the conjugate transpose of B. Show that det M_B = det B^{2n}.
2H is the set of all Hermitian matrices in V, A being Hermitian if
A=A*.Then H is a vector space over IR.Show that the function T_B
defined by T_B(A)=BAB* is a linear operator on H, and that
detT_B=det B^{2n}.
I know that the determinant of M_B is equal to that of the matrix P representing M_B .Can I find this matrix?
For part 2) I know how to show that T_B is a linear operator, and
then I guess I have to find det T_B= det M_B.... 
