Page 204 - Advanced Linear Algebra

188 Advanced Linear Algebra

(
where ÁÃÁ are the eigenvalues of . Then
²%³ ~ det ²%0 c (³
(
and setting %~ gives
det²(³ ~ c ~ ²c ³ c Ä

Hence, if is algebraically closed then, up to sign , det ² ( ³ is the constant term
-
(
of ²%³ and the product of the eigenvalues of , including multiplicity.
(
The sum of the eigenvalues of a matrix over an algebraically closed field is also
an interesting quantity. Like the determinant, this quantity is one of the
coefficients of the characteristic polynomial (up to sign) and can also be
computed directly from the entries of the matrix, without knowing the
eigenvalues explicitly.

tr
Definition The trace of a matrix ( C ²-³ , denoted by ²(³ , is the sum of
(
the elements on the main diagonal of .
Here are the basic propeties of the trace. Proof is left as an exercise.

Theorem 8.4 Let (Á )  C ²- . ³
1)tr² A³ ~ tr²(³ , for  - .
2)tr²( b )³ ~ tr²(³ b tr²)³ .
3)tr²()³ ~ tr²)(³ .
)
4 tr²()*³ ~ tr²*()³ ~ tr²)*(³ . However, tr²()*³ may not equal
tr²(*)³.
)
5 The trace is an invariant under similarity.
)
(
6 If is algebraically closed, then tr ( ² ³ is the sum of the eigenvalues of ,
-
including multiplicity, and so
tr²(³ ~ c c
c

where ²%³ ~ % b c % b Äb %b .
(
Since the trace is invariant under similarity, we can make the following
definition.
Definition The trace of a linear operator ²= ³ is the trace of any matrix
B

that represents .
As an aside, the reader who is familar with symmetric polynomials knows that
the coefficients of any polynomial
c
²%³ ~ % b c % b Äb %b

~ ²% c ³Ä²% c ³