Hassani S. Mathematical Physics: A Modern Introduction to Its Foundations
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3.4
CHANGE
OF
BASIS
AND
SIMILARITY
TRANSFORMATION
91
3.4 Change of Basis
and
Similarity Transformation
!tis
oftenadvantageous to describe aphysicalproblemiu aparticularbasis because
it takes a simpler form there, but the general form
of
the result may still be of
importance.
In
such cases the problem is solved in one basis, and the result is
transformed to other bases. Let us investigate this pointin some detail.
Given a basis
B =
(Iai)}!:!'
we can wtite an arbitrary vector la) with com-
ponents
{al,a2,
...
,aN}
in B as la) =
L!:lai
lai). Now suppose that we
change the basis to
B' =
(Iai)}f=l'
How are the components of la) in B' re-
lated to those in
B? To answer this question, we wtite lai) in terms
of
B' vectors,
lai) =
Lf=l
Pji
lai),
and substitute for lai) in this expansion of la), obtaining
la) =
L!:l
a;
Lf=l
Pji
lai)
=
Li,j
aiPji
lai)·
If
we denote the
jth
component
of
la) in
B'
by
ai,
then this equation tells us that
N
otj
= L P
jiotj
i=l
for j =
1,2,
""
N.
(3.14)
If
we use
a',
R, and a, respectively, to designate a columnvector with elements
ai,
an N x N
matrix
withelements Pit- and a columnvectorwithelements ai, then
Equation (3.14) can be wtitten in matrix form as
(
~
~
)
cp~:
~~
~~~)
(~~)
or a' = Ra.
a~
-
~l
P~2
••
a~'
(3.15)
basis
transformation
The matrixRis calledthe basis
transformation
matrix.
!t
is invertiblebecause
matrix
it is a lineartransformation that maps one basis onto another (see Theorem2.1.6).
What
happens to a matrix when we transformthe basis? Considerthe equation
Ib) = A la), where la) and Ib)have components {ail!:l and
(,Bil!:!'
respectively,
in
B. This equation has a corresponding matrix equation b = Aa. Now,
if
we
cbaoge the basis, the components of
la) and Ib) will change to those of a' and b',
respectively. We seek a matrix
A' such that b' = A'a'. This matrix will clearly be
the transform of
A. Using Equation (3.15), we wtite Rb =A'Ra, or b =R-1A'Ra.
Comparing this with b = Aa and applying the fact that both equations hold for
arbitrary
a and b, we conclude that
or
(3.16)
similarity
This is called a
similarity
transformation
on A, and A' is said to be
similar
to A.
transformation
The transformation matrix R can easily be found for orthonormal bases B =
(Jei)}!:l
and
B'
= (Ie;)}!:!. We have lei) =
Lf=l
Pki
le~).
Multiplying this
92 3.
MATRICES:
OPERATOR
REPRESENTATIONS
equation by {ei
I,
we obtain
N N
(eilei)
=
LPki
(eileiJ
=
LPki
8jk
<e».
k~1 k~1
That is,
(3.17)
3.4.1. Box.
Tofind the
ijth
element
of
the matrix that changes the compo-
nents
of
a vector in the orthonormal basis B to those
of
the same vector in
the orthonormal basis B', take the jth ket in B and multiply it by the ith bra
in R'.
To find the
ijth
element of the matrix that changes B' into B, we take the
jth
ket in
B'
and multiply it by the
ith
bra in B:
pi
j
= (ed
ei).
However, the
matrix R' must be
R-
I,
as can be seen from Equation (3.15). On the other hand,
(pi)' =
(edei)'
=
(eilei)
=Pji,Or
(R-I)lj
=Pji, or
(R-I)jj
=Pji =
(Rt)ij.
(3.18)
This shows that R is a unitary matrix and yields an important result.
3.4.2.
Theorem.
The matrix that transforms one orthonormal basis into another
is necessarily unitary.
From Equations (3.17) and (3.18) we have (Rt)ij = {ed ei}. Thus,
3.4.3. Box. To obtain the
jth
column
of
Rt, we take the
jth
vector in the
new basis and successively "multiply" it by {ed for i =
1,2,
...
, N.
In
particular,
if
the original basis is the standard basis of
eN
and lei} is rep-
resented by a column vector in that basis, then the
jth
column
of
Rt is simply the
vector
lei}.
3.4.4. Example. In this
example,
we show that the similarity transform of a function
of
a matrix is the same function
of
the similarity transform
of
the matrix:
R!(A)R-
1
=
f (RAR-
1
). Theproofinvolves
inserting
1= R-
I
Rbetween
factors
ofAinthe
Taylor
series
expansion
of
!(A):
R!(A)R-
I
= R
If.
akAk)
R-
I
=
f;akRAkR-I
=
f;akR~~-;R-I
\.t=0
k~O k~O
k times
00
.
00
=I>k
RAR-IRAR-
I
...
RAR-
I
= L ak
(RAR-I)k
=
!(RAR-
I).
III
k~O
k=O
3.5
THE
DETERMINANT
93
3.5 TheDeterminant
An important concept associated with linear operators is the determinant, Deter-
minants are usually defined in terms
of
matrix representations
of
operators in a
particular basis. This
may
give the impression that determinants are basis depen-
dent. However, we shall show thatthe value
of
the determinant
of
an operatoris the
same in all bases.
In
fact, it is possibleto define determinants
of
operators without
resort to a specific representation
of
the operator in terms
of
matrices (seeSection
25.3.1).
Letus
first
introduce
a
permutation
symbol
eiliz
..
.iN'
whichwillbeusedexten-
sively in this chapter.
It
is defined by
(3.19)
In
other words,
S'j'2
...iN is completely antisymmetric (or skew-symmetric) under
interchange
of
any pair
of
its indices. We willuse thispermutationsymbolto define
determinants. An immediate consequence
of
the definition above is that
S'j'2
...iN
will be zero
if
any two
of
its indices are equal. Also note
that
f:iliz...iN is +1 if
(iI,
ia.
...
, iN) is an even permutationI (shuflling)
of
(1, 2, . '"
N),
and
-I
if
it
is an odd permutation
of
(1,2,
...
,
N).
3.5.1 Determinantof a Matrix
determinant
defined
3.5.1. Definition. The determinant is a mapping, det :
MNxN
--> C, given in
term
of
the elements
OIu
of
a matrix A by
N
detA = L
8iliz
...
iNCXlit
...
00NiN'
il,
...
,iN=l
Definition 3.5.1 gives
detA
in terms
of
an expansion in rows, so the first entry
is from the first row, the secondfrom the secondrow, and so on.
It
is also possible
to expand in terms
of
columns, as the following theorem shows.
3.5.2. Theorem.
The determinant
of
a matrix A can be written as
N
detA = L eiriz...iNCXitl
...
otiNN·
il,
...,iN
Therefore, det A = det At.
1An even permutationmeans an even number of exchanges of pairs of elements. Thus,
(2,3,
1) is an even permutation
of (1, 2, 3), while (2, 1, 3) is an odd permutation.It can be shown (see Chapter23)
that
the parity (evenness or oddness) of
a
permutation
is well-defined; i.e., that
although
there
maybe many
routes
of
reaching
a
permutation
froma given
(fiducial)
permutation
via exchanges of
pairs
of elements, all such
routes
require
eithereven
numbers
of exchanges or odd
numbers
of
exchanges.
94 3.
MATRICES:
OPERATOR
REPRESENTATIONS
Proof
We shall go into some detail in the proofof only this theorem on determi-
nants to illustrate the manipulations involved in working with the
B symbol. We
shall not reproduce such details for the other theorems.
In
the equation of Definition 3.5.1,
ii,
i2,
...
, iN are all different and form a
permutation of
(I,
2,
...
,N).
So one
of
the a's must have I as its second index.
We assume that it is the
hth term; that is, iiI = I, and
We move this term
all the way to the left to get
aiJIali!
.
..
aNiN'
Now we look
for the entry with 2 as the second index and assume that it occurs at the
hth
position; that is, ijz = 2. We move this to the left, next to aiI1, and wtite
(ljt1(ljz2(llil
aNiN0 We
continue
inthis
fashion
untilwe get
(lhl(ljz2·
..
or
iNN.
Since
h,
n.
,
iN
is really areshuffling
of
it, i2,
...
, iN, the surnmationindices
can be changed to
h,h.
...
,l«, and we can wtite
2
N
detA =
"I:
Bili2..
.iN(1hl
...
ajNN.
il,···,jN=l
If we can show that Biliz...iN = 8
hjz
...in» we are
done.
In the
equation
of the
theorem, the sequence of integers
(ii,
i2,
...
,
is)
is obtained by some shuffling
of
(I,
2,
...
,
N).
What we have done just now is to reshuffle (iI, tz.
...
, iN) in
reverse order
to get back to (1, 2,
...
, N). Thus,
if
the shuflling in the equation
of
the theorem is even(odd), the reshuffling will also be even (odd). Thus,
Biti,
...iN =
B
iih.
..
is»
and we obtain the first part of the theorem:
N
detA = L
8hjz
...
jN(lhl
...
ajNN.
h,···,iN=l
For the second part, we simply note that the rows
of
At are coluums
of
Aand vice
versa. D
3.5.3.Theorem. Interchanging two rows (or two columns)
of
a matrix changes
the sign
of
its determinant.
Proof
The proofis a simple exercise in permutation left for the reader. D
An immediate consequence of this theorem is the following corollary.
3.5.4.Corollary. The determinant
of
a matrix with two equal rows (or two equal
columns) is zero. Therefore, one can
add
a multiple
of
a row (column) to another
row (column)
of
a matrixwithout changing its determinant.
2TheE symbolinthesumis not
independent
of thej's.
although
it
appears
without
suchindices.In
reality,
thei indicesare
"functions"
of thej
indices.
(3.20)
cofactor
of
an
element
ofa
matrix
3.5
THE
DETERMINANT
95
Since every term
of
the determinant in Definition 3.5.1 contains one and only
oneelement
from
each
row,
we canwrite
N
detA = CXilAil +cxi2Ai2 +...
+otiNAiN
=
L:CiijAij,
j~l
where
Aij
contaios products
of
elements
of
the matrix A other than the element
aii'
Sinceeachelementof aroworcolumnoccursatmostonceineachtermof the
expansion,
Aij
cannotcontainany elementfrom the
ithrow
or the
jth
column, The
quantity
Aij
is calledthe
cofactor
of
aij,
and
the above expressionis knownas the
(Laplace)
expansion
ofdetA
by its ith row. Clearly, there is a similarexpansion by
the
ith
column
of
the determinant, which is obtained by a similar argument usiog
the equation
of
Theorem 3.5.2. We collect
both
results io the followiog equation.
N N
detA =
L:cxijAij
=
L:ajiAjio
)=1 )=1
Vandermonde, Alexandre-Thieophile,alsoknownasAlexis,
Abnit,
and
Charles-Auguste
Vandermonde
(1735-1796) had a father,a physicianwho directedhis sicklyson towarda
musical
career.
An
acquaintanceship
with
Fontaine,
however,
so
stimulated
Vandennonde
that in 1771 he was elected to the Academic des Sciences, to which he presented four
mathematical
papers
(histotal mathematical
production)
in
1771
~1772.
Later,
Vandennond.e
wrote
several
papers
on
harmony,
andit was saidat thattime
that
musicians
considered
Vandermonde
tobea
mathematician
and
that
mathematicians
viewedhimasa
musician.
Vandermonde's
membership
inthe
Academy
ledto a
paper
on
experiments
withcold,
madewith
Bezout
and
Lavoisier
in 1776, anda
paper
on the
manufacture
of steel with
Berthollel
and
Monge
in 1786.
Vandermonde
becamean
ardent
and
active
revolutionary,
being such a close
friend
of Mongethathe was
termed
"femme
de
Monge."
He was a
member
of
the
Commune
of Paris andtheclub
of
the
Jacobins.
In 1782hewas
director
of
the
Conservatoire
des Artset
Metiers
andin 1792, chief of the
Bureau
de 1
'Habillement
des
Armies.
Hejoinedinthedesignof acourseinpoliticaleconomyfortheEcole
Normale
andin 1795was
named
a
member
of the
Institut
National.
Vandermonde
is best knownfor the
theory
of
determinants.
Lebesgue
believedthat
the
attribution
of
determinant
to
Vandermonde
was dueto a
misreading
of his
notation.
Nevertheless,
Vandermonde's
fourth
paper
was the
first
to give a connected exposition of
determinants,
because
he (1)
defined
a
contemporary
symbolism
that
wasmorecomplete,
simple,
and
appropriate
than
that
of
Leibniz;
(2)
defined
determinants
as
functions
apart
from
thesolution of
linear
equations
presented
by
Cramer
butalso
treated
by
Vandermonde;
and(3)gavea
number
of
properties
of these
functions,
suchasthe
number
andsignsofthe
terms
andtheeffectof
interchanging
twoconsecutive indices(rowsor
columns),
whichhe
usedto showthata
determinant
is zero
if
tworowsorcolumnsare
identical.
Vandermonde's
realand
unrecognized
claimto famewas lodgedinhis
first
paper,
in
whichhe approachedthegeneralprohlemof the solvabilityof algebraiceqoationsthrough
a
study
of
functions
invariant
under
permutations
of the rootsof the
equations.
Cauchy
96 3. MATRICES:
OPERATOR
REPRESENTATIONS
assigned priority in this to Lagrange and Vandermonde. Vandermonde read his paper in
November 1770, but he did not become a member
of
the Academy until 1771, and the
paperwasnotpublisheduntil 1774.AlthoughVandennonde'smethodswerecloseto those
laterdeveloped by Abeland Galoisfor testing the solvability
of
equations, and although his
treatment
of
the binomial equation x
n
- 1 = 0 could easily have led to the anticipation
of Gauss's results on constructible polygons, Vandennonde himself did not rigorously or
completelyestablishhisresults,nordidhe see theimplicationsforgeometry. Nevertheless,
Kroneckerdates the modern movement in algebra to Vandermonde's 1770 paper.
Unfortunately, Vandennonde's spurt
of
enthusiasm and creativity, which in two years
produced four insightful mathematical papers at least two
of
which were
of
substantial
importance, was quickly diverted by the exciting politics
of
the time and perhaps by poor
health.
3.5.5. Proposition.
Ifi
i'
k, then
L.f~1
aijAkj
=0 =
L.f~1
ajiAjk.
Proof
Consider the matrix B ohtained from Ahy replacing row k hy row i (row
i remains unchanged, of course). The matrix B has two equal rows, and its de-
terminant is therefore zero. Now, if we
ex~and
det Bby its kth row according to
Equation (3.20), we obtain0 = det B =
L.j=1
t!kjBkj.
But the elementsof the kth
row
of
Bare the elements of the
ith
row
of
A;that is, {Jkj =
ai],
and the cofactors
of
the kth row of B are the same as those
of
A, that is,
Bkj
=
Akj.
Thus, the first
equation
of
the proposition is established. The second equation can be established
using expansion by columns. D
minor
ofa
matrix
A
minor
of
order
N - I
of
an N x N matrix A is the determinant of a matrix
obtained by striking out one row and one column
of
A.
If
we strike out the
ith
row
and
jth
column of A,then the minoris denoted by
Mij.
3.5.6.
Theorem.
Aij
=
(-l)'+j
Mij.
Proof
The proofinvolves separating
all
from the rest ofthe terms inthe expansion
of the determinant, The unique coefficient of
all
is
All
by Equation (3.20). We
can show that it is also
MIl
by examining the e expansion
of
the determinant and
performing the first sum. This will establish the equality A
II
=
MIl.
The general
equality is obtained by performing enough interchanges
of
rows and columns
of
the matrix to bring
aij
into the first-row first-column position, each exchange
introducing a negative sign, thus the
(-l)i+j
factor. The details are left as an
exercise. D
The combination of Equation (3.20) and Theorem 3.5.6 gives the familiar
routine
of
evaluating the determinant of a matrix.
3.5.2 Determinants of Products of Matrices
One extremely useful property of determinants is expressed in the following the-
orem.
3.5
THE
DETERMINANT
97
3.5.7.
Theorem.
det(AB) =
(detA)(detB)
Proof
The
proof
consists
in
keeping
track
of
index
shuffling
while
rearranging
the
order
of
products
of
matrix
elements.
We
shall
leave
the
details
as an exercise. D
3.5.8. Example. Let 0 and U denote, respectively, an orthogonal and a unitary n x n
matrix; that is,
00'
=
0'0
= 1, and
UU
t
=
utu
= 1. Taking the detenoinant of the
firstequation and nsing Theorems 3.5.2 and 3.5.7. we nbtain (det O)(det
a') = (det
0)2
=
det 1 = 1. Therefore, for an orthogonal matrix, we
get
detO
=
±1.
Orthogonaltransformationspreserve a real innerproduct. Among suchtransformations
are the so-calledinversions, which, in theirsimplestform, multiply a vector
by
-1.
In
three
dimensions this correspondsto a reflectionthrough the origin. The matrix associated with
this operation is
-1:
o
-1
o
whichhas a determinant
of
-1.
This is a prototype of other, more complicated, orthogonal
transformations whose determinants are
-1.
The other orthogonal transformations, whose determinants are +I, are of special in-
terest because they correspond to rotations in three dimensions. The set
of
orthogonal
transformations in
n dimensions having the determinant
+1
is denoted by SO(n). These
transformations are special because they have the mathematical structure
of
a (continuous)
group, which finds application in many areas
of
advanced physics. We shall come back to
the topic
of
group theory later in the book.
We can obtain a similarresult for
unitary transformations. We take the determinant
of
both sides of UtU = 1:
det(U*)' det U
= det
U*
det U= (det U)*(det U) = Idet
UI
2
=
I.
Thus, we can generally write det U = e
i
a,
with a E
JR..
The
set
of
those transformations
with a = Ofonos agronp to which 1 belongs and that is denoted by SU(n). This gronphas
found applications in the attempts at unifying the fundamental interactions.
II
3.5.3 Inverse ofa Matrix
One
of
the
most
useful
properties
of
the
determinant
is
the
simple
criterion
it
gives
for
a
matrix
to
be
invertible. We
are
ready
to
investigate
this
criterion
now.
We
combine
Equation
(3.20)
and
the
content
of
Proposition
3.5.5
into
a
single
equation,
N N
LClijAkj
= (detA)8ik =
LCljiAjk,
)=1
j=l
(3.21)
98 3. MATRICES:
OPERATOR
REPRESENTATIONS
and
construct
a
matrix
CA, whoseelementsarethecofactors of theelements of the
matrix A:
(3.22)
ThenEquation (3.21) can be written as
N N
I>ij
(C~)jk
= (detA)8ik = L
(C~)kj
«s.
j=l
j=l
Of, in
matrix
form,
as
AC~
=(det A)1 =
C~A.
(3.23)
(3.24)
3.5.9.
Theorem.
The inverse
of
a matrix
(if
it exists) is unique. The matrix Ahas
inverse
ofa
matrix
an inverse
if
and only
ifdetA
i'
O.
Furthermore,
C'
A-t
=
_A_
det A'
where CA is the matrix
of
the cofactors
of
A.
Proof. Let Band C be inverses
of
A.Then
B
= (CA)B = C (AB) = C.
'-.r-'
'-.r-'
=1
=1
For
the secondpart, we note that
if
Ahas an inverse B,then
AB
= 1
=}
detAdetB
= det 1 = I,
whence det A
i'
O.
Conversely, if det A
i'
0, then dividing both sides
of
Equation
(3.23) by detA, we obtain the unique inverse (3.24)
of
A. D
The
inverse
of
a 2 x 2 matrix is easily found:
(
a
b)-t
I ( d
-b)
e d
=ad-be
-e
a
(3.25)
if
ad
- be
i'
O.
There is a more practicalway
of
calculatingthe inverse
of
matrices.
In the following discussion
of
this method, we shall confine ourselves simply to
stating a couple
of
definitions and the
main
theorem, with no attemptat providing
any proofs. The practical utility
of
the method will be illusttated by a detailed
analysis
of
examples.
3.5
THE
DETERMINANT
99
elementary
row
operation
triangular,
or
row-echelon
form
of
a
matrix
3.5.10. Definition. An elementary row operation on a matrix is one
of
the fol-
lowing:
(a) interchange
of
two rows
of
the matrix, (b) multiplication
of
a row by a
nonzero number, and
(c) addition
of
a multiple
of
one row to another.
Elementary column operations are defined analogously.
3.5.11. Definition. A matrix is in triangular, or row-echelon.form
if it satisfies
thefollowing three conditions:
1.
Any
row consisting
of
only zeros is below any row that contains at least one
nonzero element.
2.
Goingfrom
left to right, thefirst nonzero entry
of
any row is to the left
of
the
first nonzero entry
of
any lower row.
3. The first nonzero entry
of
each row is 1.
3.5.12.
Theorem.
For any invertible n x n matrix A,
• The n
x 2n matrix
(All)
can be transformed into the n x 2n matrix (IIA
-I)
by means
of
afinite number
of
elementary row operations.
3
•
if
(All)
is transformed into
(liB)
by means
of
elementary row operations,
then
B =
A-I
A systematic way of transforming
(All)
into (I
lA-I
) is first to bring A into
triangular form and then eliminate
all nonzero elements
of
each column by ele-
mentary row operations.
3.5.13.
Example.
Let us evaluate the
inverse
of
A=
G:
=D·
We start with
G
2
o I
I
-I
II
0
0)
-2
0 I 0
==
M
-I
0 0 I
o
0)
I 0
3 I
-I
II
-2
0
-5
-2
(
I 2
o I
o 0
0)
o ,
I 3(2)+(3)
o )
o
=M'
-1/5
and apply elementaryrow operations to Mto bring the left
half
of
it into triangularform.
If
we dennte the klb row by (k) and the three operatinns
of
Definition 3.5.10, respectively, by
(k)
'*
(j),
ark),
and
ark)
+
(j),
we get
(
I
2
-I
II
0
M
,0
I
-2
0 I
-2(1)+(3)
0
-3
I
-2
0
(
I
2
-I
II
0
-------+
0 I
-2
0 I
-!(3)
0 0 I
2/5
-3/5
3The matrix (AI1) denotes the n x 2n matrix obtained by juxtaposing the n x n unit matrix to the right of A.It can easily be
shown that
if
A, 8, and C are n x n matrices, then A(BIC) = (ABIAC).
100 3.
MATRICES,
OPERATOR
REPRESENTATIONS
The left half of
M'
is in
triangular
form.
However,
we wantall
entries
aboveany 1 in a
columnto be zeroas well,
.i.e.,we wantthe
left-hand
matrix
to be 1. Wecan do thisby
appropriate
use oftype3
elementary
row
operations:
c
0
3
II
-2
-tJ
M' ) 0
I
-2
0 I
-2(2)+(1)
0
0 I
2/5
-3/5
G
0
o
1-1/5
-1/5
3/5 )
,
I
-2
0 I
-~/5
-3(3)+(1)
0
I
2/5
-3/5
2(3)+(2)
G
0
o
,-1/5
-1/5
3/5 )
I
o
4/5
-1/5 -2/5
0
I 2/5
-3/5 -1/5
Theright halfof theresultingmatrixis
A-I.
11II
3.5.14. Example. It is
instructive
to
start
with a
matrix
thatis not
invertible
andshow
that
itis impossibletoturn itinto1by
elementary
row
operations.
Consider
the
matrix
B =
(i
-I
D·
-2
-I
5
Let us systematically
bring
itinto
triangular
fonn:
M=
(i
-I
31
1
0
0)
C
-2
I 10
I
D
-2
I 0 I
o
,2
-I
3 I 0
-I
5 o 0 0
I
(1)<>(2)
-I
5 o 0 0
(~I
-2
I 10
I
0)
C
-2
I 10
I
D
,
3
I I
-2
o
-------+
0 3 I I
-2
-2(1)+(2)
5 o 0 0
I
(1)+(3)
0
3 I 0 I
C
-2
I I 0
I
0)
C
-2
I I 0
I
D
' 0
3 I I
-2
o
--+
0 I
1/3 1/3
-2/3
-(2)+(3)
0
0
o
-I
3
I
~(2)
0
0
o
-I
3
The
matrix
Bis nowin
triangular
form,
butits
third
row
contains
all zeros.
There
is no
waywe can
bring
thisintotheform of a unit
matrix.
We
therefore
conclude
that
Bis not
invertible.
Thisis, of
course,
obvious,sinceit caneasily beverified thatBhasa vanishing
determinant.
11II
We mentioned earlier that the determinant is a property of linear lransfonna-
tions, although they are defined in terms
of
matrices that represent them. We can
now show this. First, note that taking the determinant of both sides
of
AA
-1
= I,