Kunze Linear Algebra · Problem 10.1.8

Bilinear Forms

Mathematics — Methods · negative-definite-bilinear-form

Problem

Let

f

be the bilinear form defined in Exercise 6, and let

V_2

be the subspace of

V

consisting of all matrices

A

such that

\text{trace}\,(A) = 0

and

A^* = -A

(

A^*

is the conjugate transpose of

A

). Denote by

f_2

the restriction of

f

V_2

. Show that

f_2

is negative definite, i.e., that

f_2(A, A) < 0

for each non-zero

A

V_2