Linear Algebra Done Right · Problem 9C.13

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · determinants

Problem

Suppose

\mathbf{F} = \mathbf{C}

T \in \mathcal{L}(V)

, and

n = \dim V > 2

. Let

\lambda_1, \ldots, \lambda_n

denote the eigenvalues of

T

, with each eigenvalue included as many times as its multiplicity. (a) Find a formula for the coefficient of

z^{n-2}

in the characteristic polynomial of

T

in terms of

\lambda_1, \ldots, \lambda_n

. (b) Find a formula for the coefficient of

z

in the characteristic polynomial of

T

in terms of

\lambda_1, \ldots, \lambda_n