Fermi
·
Concept Library
·
Courses
·
Books
·
Problems
·
Simulations
·
Math Wiki
·
High School
Fermi
›
Problems
Kunze Linear Algebra
· Problem 10.1.6
Bilinear Forms
Mathematics — Methods · bilinear-form-trace
Problem
Let
n
n
n
be a positive integer, and let
V
V
V
be the space of all
n
×
n
n \times n
n
×
n
matrices over the field of complex numbers. Show that the equation
f
(
A
,
B
)
=
n
tr
(
A
B
)
−
tr
(
A
)
tr
(
B
)
f(A, B) = n\,\text{tr}\,(AB) - \text{tr}\,(A)\,\text{tr}\,(B)
f
(
A
,
B
)
=
n
tr
(
A
B
)
−
tr
(
A
)
tr
(
B
)
defines a bilinear form
f
f
f
on
V
V
V
. Is it true that
f
(
A
,
B
)
=
f
(
B
,
A
)
f(A, B) = f(B, A)
f
(
A
,
B
)
=
f
(
B
,
A
)
for all
A
A
A
,
B
B
B
?
Browse the problem library