Exercise 19

The trace of a square matrix is the sum of its diagonal entries.

• Find the formula for the characteristic polynomial of a $2\!\times\!2$ matrix.

• Show that trace is invariant under similarity, and so we can sensibly speak of the ‘trace of a map’. (Hint: see the prior item.)

• Is trace invariant under matrix equivalence?

• Show that the trace of a map is the sum of its eigenvalues (counting multiplicities).

• Show that the trace of a nilpotent map is zero. Does the converse hold?