Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
Подождите немного. Документ загружается.
trace
ota
square
matrix
3.6
THE
TRACE
101
one obtains det(A-
1
)
=
1/
detA. Now recall thatthe representations of an opera-
tor in two different bases are related via a similarity transformation. Thus,
if
A is
represented by Ain one basis and by A' in another, then there exists an invertible
matrix R such that A'
= RAR-
1
•
Taking the determinant of both sides, we get
I
det A' =
detR
detA
--
= detA.
detR
Thus, the determinant is an inttiusic property of the operator, independent of the
basischosen in whichto
represent
the
operator.
3.6 The
Trace
Another inttiusic quantity associated with an operator that is usually defined in
terms of matrices is given in the following definition.
3.6.1. Definition.
Let
Abe an N x N matrix. The mapping tr : M
N xN
-->
<C
(or
lR)
given by tr A =
L~l
aii is calledthe trace
of
A.
3.6.2.
Theorem.
The trace is a linear mapping. Furthermore,
trA'
= trA and
tr(AB)
= tr(BA).
connection
between
trace
and
determinant
Proof
The linearity of the trace and the first identity follow directly from the
definition. To prove the second identity of the theorem, we use the definitions of
the trace and the matrix product:
N N N N N
tr(AB) =
~)AB)ii
=
LL(A)ij(B)ji
=
LL(B)ji(A)ij
;=1 ;=1
j=l
;=1
j=l
N
(N
) N
= f;
~(B)ji(A)ij
=
f;(BA)jj
=tr(BA).
D
3.6.3. Example. In this
example,
we
show
a veryuseful
connection
between
the
trace
and
thedeterminant
that
holds
when
a
matrix
is only
infinitesimally
different
from
the
unit
matrix.
Letus
calculate
the
determinant
of 1+fA to
first
order
in E. Usingthe
definition
of
determinant,
we write
n
det(1
+€A)
= L
£;1
..
.iIl(151i1
+€alit)···(l5nin
+€a'ni
n)
i1,.."i
n
= 1
n n n
= E Eit...
in81it···8nin
+€ L E
E"il
..
.in81i1···8kik
..
,8ninakik·
i1,...•i
n
= 1
k=l
;1,...,i
n
= 1
(3.26)
102 3.
MATRICES:
OPERATOR
REPRESENTATIONS
The
first
sumisjustthe
product
of all the
Kronecker
deltas.
In thesecondsum,8
kik
means
thatin the
product
of the
deltas,
~kik
is
absent.
This
term
is
obtained
by
multiplying
the
second
term
of thekth
parentheses
by the
first
term
of all therest.Since we
are
interested
onlyinthe first power
of
E, we stopat this
term.
Now,thefirst sum is reducedtoE12...n = 1
after
all the
Kronecker
deltas
are
summed
over.
Forthesecondsum,we get
n n n n
e L E Eit..
.i1l8lil···
8
kh
...
8
n
i
n
etkh
=€ L L €12 ...ik
...
n
Clki
k
k=l
it ....•
in=l
k=l
ik=l
n n
= e E€12 ...k...
netkk
= € E
akk
= € trA,
k~l
k~l
where
thelastline
follows
from
thefact
that
theonly
nonzero
value
for€12...ik...
n
is
obtained
whenik isequaltothe missing index,i.e., k,in whichcaseit will be 1.Thus
OOt(1
+fA) =
l+<trA.
III
Similar matrices have the same trace:
If
A' = RAR-
1,
then
tr
A' =tr(RAR-
1)
=tr[R(AR-
1)]
=tr[(AR-1)R]
=tr[A(R-1R)] =tr(A1) =trA.
The
preceding discussionis summarized in the following proposition.
3.6.4.
Proposition.
To every operator A E L (V) are associated two intrinsic
numbers,
det Aand tr A, which are the determinant and trace
of
the matrix repre-
sentation
of
the operator in
any
basis
of
V.
It
follows from this proposition that the resnlt
of
Example 3.6.3
can
be written
in termsof
operators:
det(l
+ fA) = 1+
<trA.
(3.27)
(3.28)
A par1icnlarly usefulformulathatcan be derivedfrom this equationis the derivative
at
t = 0
of
an operator A(t) depending on a single variable with the property
that A(O)
= 1. To first order in t, we
can
write
A(t)
= 1 +tA(o) where a dot
represents differentiating with respect to
t. Substituting this in Equation (3.27)
and differentiating with respect to
t, we obtain the important result
d I .
- det(A(t» = tr A(O).
dt
,=0
3.6.5.
Example.
Wehaveseenthatthe
determinant
ofaproductof
matrices
isthe
product
of the
determinants.
On the
other
hand,
the
trace
of asum of
matrices
is thesumof
traces.
Whendealingwith
numbers,
products
andsumsare
related
via the
logarithm
andexpo-
nential:
afJ = exp{lna +
In,B}.
A generalization of this
relation
exists fordiagonalizable
matrices.
LetAbe sucha
matrix,
i.e., let D= RAR-
1
for-some
similarity
transformation R
andsomediagonalmatrix D= diag(Al,
1.2,
...
, An).The determinantofa diagonalmatris
is simplythe
product
of its
elements:
3.7
PROBLEMS
103
Takingthenaturallog ofbothsidesandusing theresult ofExample3.2.3,wehave
In(detD)
=lnAl
+lnA2+···+lnAn
=tr(lnD),
whichcanalso be writtenas detD= exp(tr(ln D)).
In tenus ofA,thisreadsdet(RAR-
1
)
= exp{tr(ln(RAR-
1
)))
. Nowinvoketheinvariance
of
determinant
and
trace
under
similarity
transformation
andtheresultofExample3.4.4to
obtain
detA = exp{tr(R(lnA)R-
1))
= exp{tr(lnA)).
(3.29)
Thisisanimportantequation,
which
is
sometimes
usedto
define
the
determinant
of
operators
ininfinite-dimensional vector
spaces.
II
Bolhlhe determinantand lhe trace are mappings from
MNxN
to C.
The
deter-
minant is not a linear mapping,
bnt
lhe trace is; and lhis opens up lhe possibility
of
defining an innerproduct in lhe vector space
of
N x N matrices in terms
of
lhe
trace:
3.6.6.
Proposition.
For any two matrices A, B E
MNxN,
the mapping g :
MNxN
x
MNxN
--> C defined by g{A, B) =
tr(AtB)
is a sesquilinear inner
product.
Proof
The
prooffollows directly from lhe linearity
of
trace and lhe definition
of
hermitian conjugate. D
3.7 Problems
3.1. Show lhat
if
[c) = la) + Ib),lhenin any basis lhe components
of
Ie) are equal
to lhe sums
of
lhe corresponding components
of
la) and Ib). Also show lhat lhe
elements
of
lhe matrix representing lhe sum
of
two operators are lhe sums
of
the
elements
of
lhe matrices representing IDosetwo operators.
3.2. Show lhatthe unit operator 1 is represented by lhe unit matrix in any basis.
3.3.
The
linearoperator A :
]R3
-->
]R2
is given by
A
(X)
=
(2x
+Y - 3Z) .
Y
x+y-z
z
Constructlhe matrix representing A in lhe standard bases of]R3 and]R2.
3.4.
The
linear transformation T :
]R3
-->
]R3
is defined as
Find lhe matrix representation
of
T in
104 3.
MATRICES:
OPERATOR
REPRESENTATIONS
(a) the standard basis of
JR3,
(b) the basis consisting of
lat)
=
(1,1,0),
la2}
=
(1,0,
-I),
and
la3}
=
(0,2,3).
3.5. Show that the diagonal elements
of
an antisymmetric matrix are all zero.
3.6. Show that the nnmber of independent
real parameters for an N x N
1. (real) symmetric matrix is
N(N
+1)/2,
2.
(real) antisymmetric matrix is
N(N
- 1)/2,
3. (real) orthogonal matrix is
N(N
- 1)/2,
4. (complex) unitary matrix is N
2
,
5. (complex) hermitianmatrix is N
2
•
3.7. Show that an arbitrary orthogonal 2 x 2 matrix can be written in one of the
following two forms:
(
COSo
sinO
- SinO)
cos e
or
(
cosO
sinO
SinO)
-CDSe
.
The first is a pure rotation (its determinantis +
I),
and the second has determinant
-1.
The form of the choices is dictated by the assnmption that the first entry of
the matrix reduces to
I when 0 =
O.
3.8. Derive the formulas
COS(OI
+~)
= cos
01
cos~
- sin
01
sin~,
sin(OI
+~)
= sin
01
cos
~
+cos
01
sin 02
by noting that the rotation
of
the angle
01
+
~
in the xy-plane is the product
of
two rotations. (See Problem 3.7.)
3.9. Prove that
if
a matrix Msatisfies
MMt
= 0, then M=
O.
Note that in general,
M
2
= 0 does not imply that Mis zero. Find a nonzero 2 x 2 matrix whose square
is zero.
3.10. Construct the matrix representations
of
and
the derivative andmultipiication-by-t operators. Choose
II,
t, t
2
,
t
3
}
as your basis
of3'~[tl
and
{l,
t, t
2
,
t
3
,
t
4
}
as your basis of3'4[t]. Use the matrix
of
0 so obtained
to find the first, second, third, fourth, and fifth derivatives of a general polynomial
of degree 4.
3.7
PROBLEMS
105
3.11. Find the transformation matrix R that relates the (orthonormal) standard
basis
of
C
3
to the orthonormal basis obtained from the following vectors via the
Gram-Schmidt process:
Verify that Ris unitary, as expected from Theorem 3.4.2.
3.12. If the matrix representation of an endomorphism T
of
C
2
with respect to the
standard basis is
(11),
what is its matrix representation with respect to the basis
{(D,
(!I)}?
3.I3.
If the matrix representation of an endomorphism T
of
C
3
with respectto the
standard basis is
what is the representation of T with respect to the basis
o 1
-1
{( 1 ),(-1),{
I)}?
-I
1·
0
3.14. Using Definition 3.5.1, calculate the determinant of a general 3 x 3 matrix
and obtain the familiar expansion
of
such a determinant in terms of the first row
of the matrix.
3.15. Prove Corollary 3.5.4.
3.16. Showthatdet(aA)
=
aN
detAforanNxNmatrixAandacomplexnumber
a.
3.17. Show that det 1 = I for any unit matrix.
3.18. Finda specific pair
of
matrices Aand Bsuch that
det(A+
B)
oF
detA+det
B.
Therefore, the determinant is not a linearmapping. Hint:
Any
pairof matrices will
most likely work.
In
fact, the challenge is to find a pair such that det(A + B) =
detA + detB.
3.19. DemonstrateProposition3.5.5 using an arbitrary 3 x 3matrix and evaluating
the sum explicitly.
3.20. Find the inverse of the matrix
A=
(i
-2
-I
o
I
106 3. MATRICES:
OPERATOR
REPRESENTATIONS
3.21. Show explicitly that det(AB) = det Adet Bfor 2 x 2 matrices.
3.22. Given three
N x N matrices A, B,and C suchthat
AB
= C with C invertible,
show that both A and B must be invertible. Thus, any two
operatorsA and B on
a
finite-dimensional vector space satisfying AB = 1 are invertible and each is the
inverse
of
the other. Note: This is not true for infinite-dimensional vector spaces.
3.23. Show directly that the similarity transformation induced by R does not
change the determinant or the trace
of
Awhere
(
12
-1)
R = 0 1
-2
2 1
-1
and
(
3
-1
2)
A = 0 1
-2·.
I
-3
-I
3.24. Find the matrix that transforms the standard basis
of
1(;3
to the vectors
I
-I
0
.j2
.j2
!al} =
j
!a2)
=
i
!a3)
=
-2
.J6
.J6
.J6
l+i
-1+i
l+i
.J6
.J6
.J6
Show that this matrix is unitary.
3.25. Consider the three operators
LI, L2, and L3 satisfying [Lj ,
L21
= iL3,
[L3,
Ll]
= iL2. [L2, L3] =
iLl.
Show that the trace
of
each
of
these operators
is necessarilyzero.
3.26. Show that in the expansion
of
the determinant given in Definition 3.5.1, uo
two elements
of
the same row or the same columncan appear in each term
of
the
sum.
3.27. Find inverses for the following matrices using both methods discussed in
this chapter.
A=U
I
-1)
( 2
-1)
C=
(~1
-1
~
),
I 2 ,
B=
0 I
-2
,
I
-I
-2 -2
2 1
-1
-I
-2
CyO
0
(I
-
i)/(2../2)
"H)!('yO)
J
D-
0
1/../2
(I
- i)/(2../2)
-(1
+i)/(2../2)
- 1/../2
0
-(1
- i)/(2../2)
-(1
+i)/(2../2) .
0
1/../2
-(1
- i)/(2../2)
(1+
i)/(2../2)
3.28.
Let
Abe an operatoron V.Showthatif det A = 0, thenthere exists anonzero
vector
[x) E V such that A [x) =
o.
3.7
PROBLEMS
107
3.29. For which valuesofa are the following matricesiuvertible?Find the iuverses
whenever possible.
A=G
a
;),
B=G
I
D,
1
a
a
1
c=
(!
1
~),
D=G
I
~)
.
a
I
0
a
3.30.
Let
{ai}~l'
be the set consisting of the N rows of an N x N matrix Aand
assume that the
ai
are orthogonal to each other. Show that
Hint: Consider
AA
t. What would the result be
if
Awere a unitary matrix?
3.31. Prove that a set of n homogeneous linear equations in n unknowns has a
nontrivial solution
if
and only
if
the determinant of the matrix of coefficients is
zero.
3.32. Use determinants to show thatan antisymmetric matrix whose dimensionis
odd cannot have an inverse.
3.33. Show that tr(la} (bl) = (bl
a). Hint: Evaluate the trace in an orthonormal
basis.
3.34. Show that
if
two invertible N x N matrices A and B anticommute (that is,
AB
+
BA
= 0), then (a) N must be even, and (b) tr A = tr B =
O.
3.35. Show that for a spatial rotation Rfi(0)
of
an angle 0 about an arbitrary axis
n,
tr Ril
(0)
= I +2 cos O.
3.36. Express the smn of the squares of elements
of
a matrix as a trace. Show that
this smn is invariant under an orthogonal transformation
of
the matrix.
3.37.
Let
S and Abe a symmetric and an antisymmetric matrix, respectively, and
let Mbe a general matrix. Show that
(a) trM =
tr
M',
(b) tr(SA) = 0; iu particular, tr A = 0,
(c) SAis antisymmetric if and only
if
[S,
A}
= 0,
(d)
MSM'
is symmetric and
MAM'
is antisymmetric.
(e)
MHMt
is hermitian
if
His.
108 3.
MATRICES:
OPERATOR
REPRESENTATIONS
3.38. Find the trace of each of the following linear operators:
(a) T : ]R3 --> ]R3 given by
T(x, y, z) = (x +Y - z, 2x +3y - 2z, x -
y).
(b) T : ]R3 --> ]R3 given by
T(x, y, z) = (y - z,x +2y +z,z -
y).
(c) T : 1(;4--> 1(;4given by
T(x, y, z, w) = (x +
iy
- z +
iw,
2ix
+3y -
2iz
- w, x -
iy,
z +
iw).
3.39. Use Equation (3.29) to derive Equation (3.27).
3.40. Suppose that there are two operators A and B such that [A,B]
= c1, where
c is a constant. Show that the vector space in which such operators are defined
cannot
be finite-dimensional. Conclude that the position and momentum operators
of quantum mechanics can be defined only in infinite dimensions.
Additional Reading
1. Axler, S. LinearAlgebra Done Right, Springer-Verlag, 1996.
2. Birkhoff, G. and MacLane, S. Modern Algebra, 4th ed., Macmillan, 1977.
Discusses matrices from the standpoint
of
row and column operations.
3. Greub, W.LinearAlgebra, 4th ed., Springer-Verlag, 1975.
4 _
Spectral Decomposition
The last chapter discussed matrix representation
of
operators. It was pointed out
there that sucha representationis basis-dependent.
In
some bases,the operatormay
"look" quite complicated, while in others it may take a simple form. In a "spe-
cial" basis, the operator
may
look the simplest:
It
may
be a diagonal matrix. This
chapterinvestigates conditions under which a basis exists in which the operatoris
represented by a diagonal matrix.
4.1 Direct Sums
Sum
of
two
subspaces
defined
Sometimes it is possible, and convenient, to break up a vector space into special
(disjoint) subspaces. For instance, it is convenient to decompose the motion
of
a
projectile into its horizontal and vertical components. Similarly, the study
of
the
motion
of
a particle in R
3
under the influence
of
a central force field is facili-
tated by decomposing the motion into its projections onto the direction
of
angular
momentumand onto a plane perpendicularto the angularmomentum. This corre-
sponds to decomposing a vector in space into a vector, say, in the xy-plane and a
vector along the z-axis. We can generalize this to any vector space, but first some
notation: Let 11and W be subspaces
of
a vector space V. Denote by 11+W the
collection
of
all vectors in V that
can
be written as a sum
of
two vectors, one in 11
and one in W.
It
is easy to show that
11
+W is a subspace
of
V.
4.1.1.
Example.
Let
If
be the xy-plane and W the yz-plane.Theseare both subspaces
ofJll.
3,
and so is
1L
+W.
In
fact,
1L
+W =
Jll.3,
becausegivenany vector(x, y, z)
inJll.
3,
we can
write
it as
(x, y, z)
= (x, h.0) +(0, h.
z}.
'-'-'
ell
eW
110
4.
SPECTRAL
DECOMPOSITION
Thisdecomposition is not
unique:
Wecouldalsowrite(x, y, z) = (x, h,0) +(0,
~y,
z),
andahostof
other
relations.
l1li
4.1.2. Definition. Let U and W be subspaces
of
a vector space Vsuch that V=
U+
Wand
the only vector common to both U
and
W is the zero vector. Then we
direct
sum
defined
say that Vis the direct sum
ofU
and
Wand
write
uniqueness
of
direct
4.1.3. Proposition. Let U and W be subspaces
ofV.
Then V = U
Ell
W ifand only
sum
ifany vector in Vcan be written
uniquely
as a vector in
11
plus a vector in W.
Proof Assume V = U
Ell
W, and let Iv) E V be written as a sum of a vector in U
and a vector in W in two different ways:
Iv) = lu) +Iw) = lu') +Iw')
{}
lu) - lu') = Iw') - Iw).
The LHS is in U. Since it is equal to the
RHS-which
is in
W-it
must be in
W as well. Therefore, the LHS must equal zero, as must the RHS. Thus,
lu) =
lu') , Iw') = Iw), and there is only one way that [u) can be written as a sum of a
vector in U and a vector in W.
Conversely,
if
la) E U and also la) E W, then one can write
la) = la) +
10)
and
--
in'lL
inW
la) =
10)
+ la) .
--
in'lL
inW
Uniqueness of the decomposition of la) implies that la) =
10).
Therefore, the
only vector corumon to both U and W is the zero vector. This implies that V =
UEilW. D
dimensions
ina 4.1.4. Proposition.
If
V =U
Ell
W, then dim V = dim U +dim W.
direct
sum
Proof
Let llui)
}~I
be a basis for U and
Ilwi)J~=I
a basis for W. Then it is easily
verified that
IluI)
,lu2),
...
, lu
m
) ,
IWI)
,lw2),
...
,
IWk)J
is a basis for V. The
details are left as an exercise. D
Wecan generalize the notion of the direct sum to more than two subspaces. For
example, we can write
R'
=
XEIl'IJEIlZ,
where X,
'IJ,
and Z are the one-dimensional
subspaces corresponding to the three axes. Now assume that
(4.1)
i.e., V is the direct sum of r
of
its subspaces that have no common vectors among
themselves exceptthe zero vector and have the propertythat any vector in Vcan be
written (uniquely) as a sum
of
vectors one from each subspace. Define the linear
operator
Pj
by Pj lu) = lu
j)
where lu) =
LJ=I
lu
j),
lu
j)
E U
j
.
Then it is readily