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Since this holds for all vectors
u
, we must have T*(a
v
+ b
w
) = aT*(
v
) + bT*(
w
), which
proves the lemma.
Definition.
A linear transformation T :
V
Æ
V
is called
self-adjoint
if T = T*.
It follows from Lemma 1.8.3 that if a linear transformation T is self-adjoint
then
()
∑=∑
(
.
T
vwv w
T
One can prove the converse, namely
1.8.5. Lemma.
If a linear transformation T :
V
Æ
V
satisfies
()
∑=∑
(
.
T
vwv w
T
for all
v
,
w
Œ
V
, then T is self-adjoint.
Proof.
The lemma follows from the fact that for all vectors
v
we have
()
=
()
∑=∑
()
vw vwv w
∑
TT
*
.
1.8.6. Theorem.
Let
V
be a real vector space and T :
V
Æ
V
a linear transformation.
If M is the matrix of T with respect to an orthonormal basis, then the matrix of the
adjoint T* of T with respect to that same basis is M
T
.
Proof.
Let
u
1
,
u
2
,...,
u
n
be an orthonormal basis. By definition of the matrix for a
linear transformation and properties of orthonormal bases, the ijth entries of the
matrices for T and T* are T(
u
i
)·
u
j
and T*(
u
i
)·
u
j
, respectively. But
()
∑=∑
()
=
()
∑
T
uuu
T
*
u
T
*
uu
.
i
j
i
j
j
i
1.8.7. Corollary.
The matrix for a self-adjoint linear transformation on a real vector
space with respect to an orthonormal basis is symmetric. Conversely, if the matrix for
a linear transformation over a real vector space with respect to an orthonormal basis
is symmetric, then the linear transformation is self-adjoint.
Proof.
This is an easy consequence of Theorem 1.8.6.
Self-adjoint transformations are sometimes called
symmetric
transformations
because of Corollary 1.8.7. The complex analogs of Theorem 1.8.6 and Corollary 1.8.7
simply replace the transpose with the complex conjugate transpose and the self-
adjoint transformations in this case are sometimes called
Hermitian
.
We now return to the problem of when a linear transformation can be diagonal-
ized. We shall deal with real and complex vector spaces separately. The reason is that
eigenvalues are roots of the characteristic polynomial of a transformation. Although
polynomials always factor completely into linear factors over the complex numbers,
