Kunze Linear Algebra · Problem 5.4.13

Determinants

Mathematics — Methods · determinant-conjugation-operator

Problem

Let VV be the vector space of all n×nn \times n matrices over the field of complex numbers, and let BB be a fixed n×nn \times n matrix over C\mathbb{C}. Define a linear operator MBM_B on VV by MB(A)=BAB∗M_B(A) = BAB^*, where B∗=Bt‾B^* = \overline{B^t}. Show that det⁡MB=∣det⁡B∣2n.\det M_B = |\det B|^{2n}. Now let HH be the set of all Hermitian matrices in VV, AA being Hermitian if A=A∗A = A^*. Then HH is a vector space over the field of real numbers. Show that the function TBT_B defined by TB(A)=BAB∗T_B(A) = BAB^* is a linear operator on the real vector space HH, and then show that det⁡TB=∣det⁡B∣2n\det T_B = |\det B|^{2n}. (Hint: In computing det⁡TB\det T_B, show that VV has a basis consisting of Hermitian matrices and then show that det⁡TB=det⁡MB\det T_B = \det M_B.)

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