 Solution

ID: past-exam-of-the-algebra-1-and-differential-equations-1-course-of-the-university-of-oxford/2014/2/c/i/solution

Rearrange

M^{T} J M = J

M^{- 1} = J^{- 1} M^{T} J .

Thus

M^{- 1}

is similar to

M^{T}

. Similar matrices have the same characteristic polynomial by part (a), and transposition also preserves it:

det (x I - M^{T}) = det ((x I - M)^{T}) = det (x I - M) .

Therefore

χ_{M^{- 1}} = χ_{M}

. This is the underlying algebraic source of the reciprocal spectrum of a symplectic matrix.

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