Linear operator and determinant

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

Hi, I am having trouble with the following exercise.

1-Let 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}.

2-H 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....

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