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satisfying 〖(X)bar〗^t = -X (skew-hermitian).

Consider M as a vector space over R.

Define a bilinear form B on M by B(X,Y) = -tr(XY)

(1) Show that B takes real values, is symmetric and positive definite.

(2) For any A ∈ M , define the operator ad_A: M → M by ad_A(X) = AX – XA.

Show that operator ad_A is diagonalizable.

(3) Let the matrix

A =

( i 1)

(-1 i) .

Compute the eigenvalues of operator ad_A.

(For part (2), Maybe we should show there is a basis of M consisting of eigenvectors of ad_A?)

Thanks.