Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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2.7
PROBLEMS
81
2.41. Find an expression for J
a
in powers
of
a. Retain all terms np to the foorth
power.
2.42. Show that for a = 2, the third power
of
a disappears in Equation (2.29).
2.43. Evaluate the following integrals numerically, using six subintervals with
the trapezoidal rule, Simpson's one-third rule, and Simpson's three-eighths rule.
Compare with the exact result when possible.
1
5
x
3dx.
[,
f2
(a) (b)
o
e-
X
dx.
(c) 0
xei
cosx
dx.
[1
14inXdx.
{'
(d)
-dx.
(e)
(f)
1
e'
sin x
dx.
1 x
[ dx
[,
(i)
1
1
eX
tan x dx.
(g)
_1
1
+
x2
'
(h) 0
xe"
dx.
Additional Reading
1. Axler, S. Linear Algebra Done Right, Springer-Verlag, 1996.
2. Greub, W.
LinearAlgebra,
4th
ed., Springer-Verlag, 1975.
3. Hildebrand,
F.Introduction to Numerical Analysis,
2nd
ed., Dover, 1987.
Uses operatortechniques in numerical analysis.
It
has a detailed discussion
of
error analysis, a topic completely ignored in our text.
3 _
Matrices:
Operator
Representations
So far, our theoretical investigation has been dealing mostly with abstract vec-
tors and abstract operators. As we have seen in examples and problems, concrete
representations of vectors and operators are necessary in most applications. Such
representations are obtained by choosing a basis and expressing
alloperations in
terms of components of vectors and matrix representations of operators.
3.1 Matrices
Letus
choosea basis Bv = {Iai)
1~1
ofa
vector space VN, and express an arbitrary
vector Ix) in this basis: [r) =
L~I
~i
lai). We write
(3.1)
representation
of and say that the column vector x represents Ix) in
Bv
. We can also have a linear
vectors
transformation A E L(VN, WM) act on the basis vectors in Bv to give vectors in
the M-dimensional vector space WM:
IWk)
= A lak). The tatter can be written as
a linear combination of basis vectors Bw = {Ibj)
If=j
in W
M
:
M
IWI)
=
L"jllbj),
j=l
M
IW2)
=
L"j2Ibj),
j=l
M
IWN)
=
L"jN
Ibj)'
j=1
Note that the components have an extra subscriptto denote which
of
the N vectors
{IWi )
1~1
they are representing. The components can be arranged in a column as
3.1
MATRICES
83
before to give a representation of lbe corresponding vectors:
The operator itself is determined by the collection of all these vectors, i.e., by a
matrix. We write this as
(
"'11
"'21
A=
.;
a~2
(3.2)
representation
of
operators
and call Albe
matrix
representing
Ain bases Bv
and
Bw. This statementis also
sunnnarized symbolically as
M
Alai) =
I>jilbj)
,
j~1
i = 1,2,
...
,N.
(3.3)
We lbus have the following mle:
3.1.1.Box. To find the matrix Arepresenting A in bases Bv = (Iai)
1~1
and Bw =
lib
j)
1f=1'
express
Alai)
as a linear combination
of
the vectors
in
Bw. The components form the ith column
of
A.
Now consider lbe vector Iy) = AIx) in WM. This vector can be written in two
ways: On the one hand,
Iy) =
Ef=1
ryj Ib
j
). On the other hand,
N N
IY)
=Alx)
= A
I)
lai) =
L~iAlai)
;=1
;=1
Since Iy) has a unique set of components in lbe basis
Bw,
we conclude lbat
N
ryj =
L"'ji~i'
i=l
j = 1,2,
...
,M.
(3.4)
84 3. MATRICES:
OPERATOR
REPRESENTATIONS
Thisis
written
as
(
m)
(al1
~2
a21
· .
·
.
· .
~M
aMI
a12
a22
aM2
(3.5)
The
operator
T
A
associated
with
a
matrix
A
in which the matrix multiplication rule is understood. This matrix equation is the
representation
of
the operatorequation Iy) = A [x) in the bases
Bv
and
Bw.
The construction above indicates
that-once
the bases are fixed in the two
vector
spaces-to
every operator there corresponds a unique matrix. This unique-
ness is the result
of
the uniqueness
of
the components
of
vectors in a basis. On
the other hand, given an
M x N matrix A with elements
aij,
one
can
construct a
unique linear
2Perator
TA definedby its action on the basis vectors (see
Box
1.3.3):
TA lai)
==
Ljd
aji
Ib
j
).
Thus, there is a one-to-onecorrespondence between op-
erators and matrices. This correspondence is in fact a linearisomorphism:
3.1.2.
Proposition.
The two vector spaces £.,(V
N,
W
M)
and
JV(M xN are isomor-
phic. An explicit isomorphism is established only when a basis is chosen for each
vectorspace,inwhichcase,anoperator isidentifiedwithitsmatrixrepresentation.
Given the linear transformations A : V
N
-+
WM and B : WM
-+
UK,
we can form the composite linear transformation
BoA
: VN
-+
UK. We
can
also choose bases
Bv
=
{Iai)}~"
Bw =
{Ibi)}~"
Bu
= {Ici)};;', for V, W,
and U, respectively. Then A, B, and
BoA
will be represented by an M x N, a
K x M, and a K x N matrix, respectively, the latter being the matrix product
of
the other two matrices. Matrices are determined entirely by their elements. For
this
reason
a
matrix
A whoseelements are
(.lll,
0!12,
...
is
sometimes
denoted
by
(aij). Sintilarly, the elements
of
this matrix are denoted by (A)ij. So, on the one
hand, we have
(aij)
=A, and on the other hand (A)ij =
ai],
In the context
of
this
notation,
therefore,
we can
write
(A+B)ij =(A)ij +(B)ij
=}
(aij
+fJij) =
(aij)
+(fJij),
(yA)ij
=
y(A)ij
=}
y(aij)
=
(yaij),
(O)ij = 0,
(1)ij = 8ij.
A matrix as a representation
of
a linear operator is well-defined only in refer-
ence to a specific basis. A collection
of
rows and columns
of
numbersby themselves
have no operational meaning. When we manipulate matrices and attach meaning
to them, we make an unannounced assumption regarding the basis: We have the
standard basis
of
en
(or
JRn)
in mind. The following example should clarify this
subtlety.
3.1
MATRICES
85
3.1.3.
Example.
Let us fiud the matrix represeutatiou of the liuear operator A E £,
(lR.
3),
given by
(
X)
(X-Y+2Z)
A Y =
3x-z
z
2y+z
inthebasis
There
is a
tendency
toassociate the
matrix
-1
o
2
(3.6)
withthe
operator
A.Thefollowingdiscussionwill showthatthisis false.To
obtain
the
first
columnof the
matrix
representing
A,we consider
So, by Box 3.1.1, the first column of the matrix is
Theothertwo
columns
are
obtained
from
givingthesecondandthe
third
columns,
respectively. Thewhole
matrix
is then
2
-5~)
1 t .
o 2
Aslongasall
vectors
are
represented
bycolumnswhose
entries
are
expansion
coefficients
of the
vectors
in B, A andA
are
indistinguishable.
However,
theactionof Aonthecolumn
86 3.
MATRICES:
OPERATOR
REPRESENTATIONS
x
vector
(Y
)willnotyieldtheRHSofEquation(3.6)!Althoughthisisnotnsuallyemphasized,
z
thecolumnvectorontheLHSof Equation (3.6) is reallythevector
whichis an
expansion
in
terms
of thestandard basis
of]R3
rather
than
in tenusof B.
x
Wecan
expand
A(Y) in
terms
of B. yielding
z
(
X)
(X
-Y +
2Z)
A Y =
3x-z
z
2y+z
= (2x - b)
(i)
+(-x+
h+
2z)
G)
+(x+ b
-z)
(D.
Thissays
that
in thebasisB this
vector
hasthe
representation
X
Similarly,
(Y
)is represented by
z
W))
= ( t
~
~~:
~;
).
z B
-~x
+
~y
+
~z
ApplyingA to theRHSof (3.8)yieldsthe RHSof (3.7),as it should.
(3.7)
(3.8)
Given any
M X N matrix A, an operatorTA E L (V
N,
W
M)
can be associated
with A, and one can construct the kernel and the range
of
TA. The rank
of
TA
rank
ofa
matrix
is called the rank
of
A. Sioce the rank
of
an operator is basis independent, this
definition makes sense.
Now suppose that we choosea basisfor the kernel
ofT
A and extend it to a basis
of
V.Let VI denote the span
of
the remainingbasis vectors. Similarly, we choose a
basis for T
A (V) and extendit to a basisfor W.
In
these two bases, the M x N matrix
representiog T
A will have all zeros except for an r x r submatrix, where r is the
rank
of
T
A.
The
reader
may
verify that this submatrix has a nonzero detenninant.
In
fact, the submatrix represents the isomorphism betweeu VI and TA (V), and,
by its very construction, is the largest such matrix. Since the determinant
of
an
operator is basis-iudependent, we have the followiog proposition:
3.1.4.Proposition. The rank
of
a matrix is the dimension
of
the largest (square)
submatrix whosedeterminant is notzero.
3.2
OPERATIONS
ON
MATRICES
87
3.2 Operations on Matrices
There are two basic operations that one
can
perform on a matrix to obtain a new
transpose ofa
matrix
one; these are transposition
and
complexconjugation.
The
transpose
of
an M x N
matrix A is an N x M matrix At obtained by interchanging the rows and columns
of
A:
(At)ij = (A)
ji,
or
(3.9)
The
following theorem, whose
proof
follows immediately from the definition
of
transpose, summarizesthe importantproperties
of
the operation
of
transposition.
3.2.1.
Theorem.
Let Aand Bbe two (square) matrices. Then
(a) (A+B)t = At +B
t,
(b) (AB)t = B
t
At,
(c) (At)t = A.
(3.10)
symmetric
and
antisymmetric
matrices
orthogonal
matrix
complex
conjugation
hermitian
conjugate
Of
special interest is a matrix that is identical to its transpose.
Such
matrices
occur frequently in physics
and
are called
symmetric
matrices. Similarly,
anti-
symmetric
matrices are those satisfying At =
-A.
Any matrix A
can
be written
as A =
~(A
+At) +
~(A
- At), where the first term is symmetric
and
the second
is antisymmetric.
The
elements
of
a symmetric matrix A satisfy the relation a ji = (At)ij =
(A)ij =
aij;
i.e., the matrix is symmetric
under
reflection through the main di-
agonal. On the other hand, for an antisymmetric matrix we have
a ji =
-aij.
In
particular, the diagonal elements
of
an antisymmetric matrix are all zero.
A (real) matrix satisfying
AtA =AA
t
= 1 is called
orthogonal.
Complex
conjugation
is an operation
under
which all elements
of
a matrix
are complex conjugated. Denoting the complex conjugate
of
A by A*, we have
(A·)ij
=
(A)i
j,
or (aij)*
==
(a~).
A matrix is real
if
and only
if
A* = A. Clearly,
Wr=A .
Under
the combined operation
of
complex coujugation
and
transpositiou, the
rows and columns
of
a matrix are interchangedand all
of
its elemeuts
are
complex
conjugated. This combinedoperatiouis calledthe
adjoint
operation,or
hermitian
conjugation,
and is denoted by
t,
as with operators. Thus, we have
At
= (A
t)*
= (A*)t,
(At)ij
= (A)ji or
(aij)t
=
(aji)'
Two types
of
matrices are important enough to warrant a separate definition.
hermitian
and
unitary
3.2.2.Definition. A hermitianmatrix Hsatisfies Ht = H,or, in terms
of
elements,
matrices
~ij
=tl
ji-
A unitary matrix U satisfies
UtU
=uut=1, or, in terms
of
elements,
"N
*"N*
,
L....k=l
lLiktL
jk
=
L...k=l
J.Lkif.Lkj =
Vij·
Remarks:
It
follows immediately from this definition that
I.
The
diagonal elements
of
a hermitian matrix are real.
88 3.
MATRICES:
OPERATOR
REPRESENTATIONS
2. The kth column
of
a hennitiau matrix is the complex coujugate
of
its kth row,
and
vice
versa.
3. A real hermitiau matrix is symmetric.
4. The rows
of
au N x N unitary matrix, when considered as vectors in eN, form
au orthonormal set, as do the columns.
5.
A real unitary matrix is orthogonal.
It
is sometimes possible (aud desirable) to trausfonn a matrix into a form in
which all
of
its off-diagonal elements are zero. Such a matrix is called a diagonal
diagonal
matrices
matrix. A diagonal matrix whose diagonal elements are
{Akl~1
is denoted by
diag(AI, A2,
...
,
AN)'
3.2.3. Example. In this example,we derivea usefulidentityfor functions of a diagonal
matrix.Let D
= diag(AI,
1.2,
...
, An) be adiagonalmatrix,and
[(x)
a functionthat
bas
a
Taylor
series
expansion
f (x) =
L~o
akxk. Thesame
function
of Dcanbe
written
as
In
words,
the
function
ofa
diagonal
matrix
is
equal
toa
diagonal
matrix
whose
entries
are
the
same
function
of the
corresponding
entries
of the
original
matrix.
ill theabove
derivation,
we usedthefollowingobvious
properties
of
diagonal
matrices:
adiag(AJ,A2,
...
,An)
=diag(aAI,aA2,
...
,aAn),
diag(AI,
1.2,
, An) + diag("'I, "'2, ,
"'n)
=
diag(AI
+"'1,
...
,
An
+ Wn),
diag(AI,
1.2,
, An) . diag("'I. "'2 ,
"'n)
= diag(AI"'I,
...
, An"'n).
III
3.2.4. Example. (a) A prototypical symmetric matrix is that of the moment of in-
ertia
encountered
in
mechanics.
The
ijth
eleIIlent
of this
matrix
is
defined
as
Iij
es
fff
P(XI.
X2, X3)XiXj
dV,
whereXi is the ilb Cartesiancoordinateof a pointin the dis-
tribution
of mass
described
bythevolume
density
P(Xlo
xz, X3).It is clear
that
Iij
=
Iji.
orI = It. The
moment
of
inertia
matrix
canbe
represented
as
Ithassix
independent
elements.
(b) Anexampleof anantisymmetric
matrix
is the
electromagnetic
field
tensor
givenby
F=
(~3
-82
-EI
-83
82
o
-81
81 0
-E2 -E3
Pauli
spin
matrices
(c)
Examples
of hermitian
matrices
arethe2 x 2 Pauli spin matrices:
3.3
ORTHONORMAL
BASES
89
(
0
-i)
cz
= i 0 .
(d)Themost
frequently
encountered
orthogonal
matrices
are
rotations.
Onesuch
matrix
Euler
angles
represents
the
rotation
ofa
3-dimensional
rigidbodyin
terms
of Euleranglesandisusedin
mechanics.
Attaching
a
coordinate
system
tothe
body,
a
general
rotation
canbe
decomposed
intoa
rotation
of anglee
about
thez-axis,followedbya
rotation
of anglee
about
thenew
x-axis, followedbya
rotation
ofangle
1ft
about
thenew z-axis. We
simply
exhibit
this
matrix
in
terms
of these
angles
andleaveittothe
reader
to show
that
itis indeed
orthogonal.
(
COS
1fr
cos tp - sin
1/1
cos 8 sin
cp
sin
1/1
cos
rp
+cos
'if.r
cos ()sin
rp
sin
fJ
sin
rp
- cos
1fr
sin
q;
- sin
1/1
cos 8 cos
rp
- sin
""
sin
cp
+cos
1/1
cos ()cos qJ
sine
cos
rp
sin1f.rsin9
)
- cos
1ft
sinB .
cosB
III
3.3 Orthonormal Bases
Matrix representation is facilitated by choosing an orthonormal basis B =
(Ie;)}!:!.
The matrix elements
of
an operator A can be found in such a basis
by "multiplying" both sides of
A lei) =
I:f=t
aki lek) on the left by {ej
I:
or
(3.11)
We can also show that in an orthonormal basis, the ith component
;i
of a
vector is found by multiplying it by
(ei
I.
This expression
for;i
allows us to write
the expansion of [x) as
N
=}
1 =
I>j}(ejl,
j~l
(3.12)
which is the same as in Proposition2.5.3. Let us now investigatethe representation
of the special operators discussed in Chapter 2 and find the connection between
those operators and the matrices encountered in the last section. We begin by
calculating the matrix representing the hermitian conjugate of an operator
T. In
an orthonormal basis, the elements of this matrix are given by Equation (3.11),
7:;j
= (eil T lej). Taking the complex conjugate of this equation, and using the
definition of
T" given in Equation (2.8), we obtain
7:
i
j =(eil Tlej)* =(ej Irt lei),
or (Tt)ij
=
7:j;.
This is precisely how the adjoint
of
a matrix was defined.
90 3. MATRICES:
OPERATOR
REPRESENTATIONS
Note how crucially this conclusion depends on the orthonormality of the basis
vectors.
If
the basis were not orthonormal, we could not use Equation (3.11) on
which the conclusion is based. Therefore,
3.3.1. Box. Only in an orthonormalbasis is the
adjoint
of
an operatorrep-
resented by the adjoint
of
the
matrix
representing that operator.
In particular, a hermitian operator is represented by a hermitianmatrix only
if
an orthonormal basis is used. The following example illustrates this point.
3.3.2.
Example.
Consider
the
matrix
representation
of the
hermitian
operator
Hin a
general-notorthononnal-basis
B = {Iai) If::l.Theelements
of
the
matrix
corresponding
toH aregivenby
N
Hlak) = L tl
jk
laj).
j=l
or
N
H la;) = L tl
ji
laj}'
j=l
(3.13)
Takingthe product of the firstequation with
<rti
Iand complex-conjugatingthe result gives
(aiIHlak)* =
CLf~l~jk(ailaj})*
=
Ej=l~jk(ajlai).Butbythedefinitionofa
hermitian
operator,
(ailH lak)* = (aklHt lai) = (aklH lai)'
So we have (aklH lai) =
Ef=l
~jk
(ajlai).
On the other hand, muitiplying the second equation in (3.13) by (akl gives
(aklHjai) =
"L.7=1
Ylji {akl
aj}'
Theonlyconclusion we can
draw
from
this
discussion
is
Ef~l
~jk
(ajl ail =
Ef~l
~ji
(akl
aj).
Because litis equation does not say anything
about
each
individual
1Jij. we cannotconclude, in
general,
that
1]0
=
'YJji'
However,
if the lai)'s are orthonormal, then
(ajl
ail = 8ji and (akl
aj)
= 8kj. and we obtain
"L.f=l11jk(Jji
= "Lf=11Jji8icj, or
rJik
=
nu.
as
expected
of ahermitian
matrix.
II
Similarly, we expect the matrices representing unitary operators to be uni-
tary only
if
the basis is orthonormal. This is an inunediate consequence of
Equation (3.10), hut we shall prove it in order to provide yet another example
of
how the completeness relation, Equation (3.12), is used. Since
UU
t
= 1,
we have
(eil
UU
t
lej)
=
(eil1Iej)
= 8ij. We insert the completeness relation
1
= Lf=llek) (ekl between Uand uton the LHS:
(eil U
(~
lekHekl)
U
t
lej)
=
~
(eil
~
lek) (ekl
~t
lej)
= 8ij.
=f.£ik =f.£jk
This equation gives the first half
of
the requirement for a unitary matrix given in
Definition 3.2.2. By redoing the calcnlation for UtU, we conld obtain the second
half
of
that requirement.