16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 10.4.9

Bilinear Forms

Mathematics — Linear Algebra · invariant-bilinear-form-orthogonal

Problem

Let

V

be the space of all

n \times 1

matrices over

C

A

n \times n

matrix over

C

, and

f

the bilinear form on

V

given by

f(X, Y) = X^t A Y

. Show that

f

is invariant under

O(n, C)

, i.e.,

f(MX, MY) = f(X, Y)

for all

X, Y

V

and

M

O(n, C)

, if and only if

A

commutes with each member of

O(n, C)