 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/18/c/solution

The expression assumes

Y

Tr (Y X Y^{- 1}) = Tr (X Y^{- 1} Y) = Tr X .

For a real or complex antisymmetric matrix,

X^{T} = - X

forces every diagonal entry to satisfy

x_{ii} = - x_{ii}

, hence

x_{ii} = 0

. Therefore

Tr Z = Tr X = 0.

This also demonstrates invariance of the matrix trace under matrix similarity. Antisymmetry of

Z

itself is not needed.

 New to topics? Read the docs here!