16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.2.10

Bilinear Forms

Mathematics — Linear Algebra · hermitian-matrices-quadratic-form

Problem

Let

V

be the real vector space of all

2 \times 2

(complex) Hermitian matrices, that is,

2 \times 2

complex matrices

A

which satisfy

A_{ij} = \overline{A_{ji}}

. (a) Show that the equation

q(A) = \det A

defines a quadratic form

q

V

. (b) Let

W

be the subspace of

V

of matrices of trace

0

. Show that the bilinear form

f

determined by

q

is negative definite on the subspace

W