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Kunze Linear Algebra
· Problem 8.4.2
Inner Product Spaces
Mathematics — Methods · unitary-operators-matrices
Problem
Let
V
V
V
be the space of complex
n
×
n
n \times n
n
×
n
matrices with inner product
(
A
∣
B
)
=
tr
(
A
B
∗
)
(A|B) = \operatorname{tr}(AB^*)
(
A
∣
B
)
=
tr
(
A
B
∗
)
. For each
M
M
M
in
V
V
V
, let
T
M
T_M
T
M
be the linear operator defined by
T
M
(
A
)
=
M
A
T_M(A) = MA
T
M
(
A
)
=
M
A
. Show that
T
M
T_M
T
M
is unitary if and only if
M
M
M
is a unitary matrix.
Related theory
Hermitian Skew Hermitian And Unitary Matrices
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