Upload
satin
View
38
Download
0
Embed Size (px)
DESCRIPTION
Chapter 3 Determinants and Matrices. September 14 Determinants 3.1 Determinants. Fundamental definition :. Notes for each term in det A : Each row has only one number. Each column has only one number. The sign is given by the permutation. Levi-Civita symbol :. - PowerPoint PPT Presentation
Citation preview
Fundamental definition:
1
assumed) over (summation det
,matrix an For
321 i,j,k,aaa
ann
kjiijk
ij
A
A
September 14 Determinants
3.1 Determinants
Chapter 3 Determinants and Matrices
backward. movie Play the :Proof
).( to)(123 from as same theis )(123 to)( fromn permutatio ofnumber The
equal. are indices any two if ,0
).(123 ofn permutatio oddan is )( if ,1
).(123 ofn permutatioeven an is )( if ,1
ijkijk
ijk
ijk
ijk
number of interchanges of adjacent elements
Levi-Civita symbol:
Notes for each term in det A:1)Each row has only one number.2)Each column has only one number.3)The sign is given by the permutation.
2
AA
A
AA
det ~
det
)()123(
)()123()123()(
Then . isThat numbers.adjacent
twoinginterchangby into change tosteps need weSuppose
. Suppose
~~~~det
:Proof
det ~
det
3213'2'1''''
3213'2'1'
'''steps
steps steps
321
steps
3'2'1'
3213'2'1'
3213'2'1'
3'2'1'''''3'2'1'''
kjiijkkjikji
kjikji
ijkkjim
mm
kji
m
kji
kjikji
kjikji
kjikjikjikji
aaaaaa
aaaaaa
ijk
i'j'k'i'j'k'
aaaaaa
aaaaaam
aaaaaa
aaaaaa
3
Development by minors (an iterative procedure):Minor A(i,j): A reduced array from A by removing the ith row and the jth column.
aa
a
a
i
jiij
ji
j
jiij
ji
det
det)1(
det)1(det
),(
),(
A
AA
Cofactor Cij: ),(det)1( jiji
ijC A
Expanding along a row
Expanding along a column
i
ijijj
ijij CaCaAdet
Useful properties of determinants:
1)A common factor in a row (column) may be factored out.2)Interchanging two rows (columns) changes the sign of the determinant.3)A multiple of one row (column) can be added to another row (column) without changing the determinant.These properties can be tested in the triple product of
321
321
321
)()()(
CCC
BBB
AAA
ACBBACCBA
4
Homogeneous linear equations:
The determinant of the coefficient matrix must be zero for a nontrivial solution to exist.
.0 if 00
0
0
0
000
333231
232221
131211
1
3332
2322
1312
3332333232131
2322323222121
1312313212111
3332131
2322121
1312111
333231
232221
131211
1
333232131
323222121
313212111
aaa
aaa
aaa
x
aa
aa
aa
aaxaxaxa
aaxaxaxa
aaxaxaxa
aaxa
aaxa
aaxa
aaa
aaa
aaa
xxaxaxaxaxaxaxaxaxa
Inhomogeneous linear equations:
. and for solutionssimilar and , 32
333231
232221
131211
33323
23222
13121
1
33323
23222
13121
3332333232131
2322323222121
1312313212111
3332131
2322121
1312111
333231
232221
131211
1
3333232131
2323222121
1313212111
xx
aaa
aaa
aaa
aac
aac
aac
x
aac
aac
aac
aaxaxaxa
aaxaxaxa
aaxaxaxa
aaxa
aaxa
aaxa
aaa
aaa
aaa
x
cxaxaxa
cxaxaxa
cxaxaxa
5
Linear independence of vectors:
A set of vectors are linearly independent if the only solution for is
n,,, aaa 21
02211 nnxxx aaa .021 nxxx
Gram-Schmidt orthogonalization:
Starting from n linearly independent vectors we can construct an orthonormal basis set
,21 n,,, vv v .21 n,,, ww w
.
,
,
,
1
1
1
1
2231133
22311333
1122
11222
1
11
k
iiikk
k
iiikk
k
wwvv
wwvvw
wwvwwvv
wwvwwvvw
wwvv
wwvvw
v
vw
6
Read: Chapter 3: 1Homework: 3.1.1,3.1.2Due: September 23
7
September 16,19 Matrices
3.2 Matrices
Definition: A matrix is a rectangular array of numbers.Terminology: row, column, element (entry), dimension, row vector, column vector.
Basic operations:
Addition:
Scalar multiplication:
Transpose:
ijijij ba BA
ijij cac A
TTTTjiijij aa ABABAA )( ), (sometimes ~~
Rank: The maximal number of linearly independent row (or column) vectors is called the row (or column) rank of the matrix. For any matrix, row rank equals column rank.Proof (need more labor): 1) Elementary row operations do not change the row rank. 2) Elementary row operations do not change the column rank. 3) Elementary row operations result in an echelon form of matrix, which has equal row and column ranks.
Elementary row (column) operations:1)Row switching.2)Row multiplication by a number.3)Adding a multiple of a row to another row.
8
Matrix multiplication: . then If kjikij bac ABC
. Usually 3). ,)()(2)
match. should dimentions The 1)
BAABACABC)A(BCABBC A
In the view of row (or column) vectors:
jkjkj
jjijjijiji
a
aabac
bcbc
bcbc
CAB
.) vectorsrow are and (
)()(
then, If
1
111111
The kth row of C is a linear combination of all rows of B, each weighted by an element from the kth row of A.
(Similarly by taking the transpose)The kth column of C is a linear combination of all columns of A, each weighted by an element from the kth column of B.
CBA
9
Product theorem: BAAB detdet )det(
BA
b
b
b
b
b
b
b
b
b
b
b
b
C
b
b
b
c
c
c
C
A
BCbc
bcbcbc
ABC
detdetdet
detdetdetdet
.) of rowth thefromelement an by
tedeach weigh , of rows all ofn combinatiolinear a is of rowth (The
vectors.)row are and ( )()(
.Let :Proof
2
1
21
21
2
1
2
1
2
1
2
1
1111111
2121
2
1
21
22
1
1
22
11
22
11
n
jjjnjjj
j
j
j
njjj
jnj
jj
j
j
jnj
jj
jj
jnj
jj
jj
n
jkjk
jjjijjijiji
nn
n
n
nnnn
nn
kk
aaa
aaa
a
aa
a
a
a
a
a
a
k
ka
aabac
)()(
)()(
DCBDCA
DCBA
ba
ba
10
Direct product:
BBB
BBB
BBB
BA
mnmm
n
aaa
aaa
aaa
21
212221
11211
. is ofdimension then the, is , is If
nqmpqpnm
BABA
Diagonal matrices:
. then matrices, diagonalboth are and If
00
00
00
22
11
BAABBA
A
nna
a
a
Trace:
)(Tr)(Tr)(Tr
)(Tr)(Tr
)(Tr)(Tr)(Tr
)~
(Tr)(Tr )(Tr
CABBCAABC
BAAB
BABA
AAA
ijijji
ijjiij
iii
abba
a
11
Matrix inversion:
AA
A
ABAB
AAAA
det
~
det ,
)(
1
)1()1(1
111
11
ijjiijij
CCaa
Gauss-Jordan method of matrix inversion:
Let MLA=1 be the result of a series of elementary row operations on A, then . 1 1 AMM LL
Example:
.10
01
2/12/3
12
43
21
:Test
2/12/3
12
10
01
13
12
20
01
13
01
20
21
10
01
43
21
)2/1()2(
)2()1(3)1()2(
12
Read: Chapter 3: 2Homework: 3.2.1,3.2.31,3.2.34,3.2.36Due: September 23
13
September 21 Orthogonal matrices
3.3 Orthogonal matrices
Change of basis (rotation of coordinate axes):
change coordinate , 'ly Particular
' )ˆ'ˆ('ˆ'ˆ'
'ˆ'ˆ)ˆ'ˆ(ˆ
ˆ'ˆ ˆˆ)ˆ'ˆ('ˆ
Arr'
eeeeV
eeeee
eeeeeee
jiji
jijijijijjii
jjijiji
jiijjijjjii
xax
VaVVVVV
a
aa
Orthogonal transformation: (orthonormal transformation) preserves the inner product between vectors:
2121 '' VVVV
jkikijikijkjkikjijiiiijiji aaaaxxxaxaxxxxxax '' ,'For
Orthogonality conditions:
Other equivalent forms:
jkkijiaa
1~
~1
~
1
AA
AA
AA
14
Orthogonal matrix: 1~ AA
• An orthogonal matrix preserves the inner product of vectors.
• The determinant of any orthogonal matrix is +1 (rotation) or −1 (rotation + inversion).
• All 3 ×3 orthogonal matrices form an orthogonal group O(3). Its subgroup SO(3) (special orthogonal group) consists of the orthogonal matrices with determinant +1.
Similarity transformation:
The matrix representation of an operator depends on the choice of basis vectors.
Let operator A rotate a vector:
B change the basis (coordinate transformation):
Question:
basis) new(in ''' basis); old(in 11 rArArr
.' ,' 11 BrrBrr
111 ' '''''' BABABABABrABArBrABrrAr
A′ and A are called similar matrices. They are the representation of the same operator in different bases.
AAAA
Tr'Trdet'det
?),(' BAA f
15
Read: Chapter 3: 3Homework: 3.3.1,3.3.8,3.3.9,3.3.10,3.3.14Due: September 30
16
September 23,26 Diagonalization of matrices
3.4 Hermitian matrices and unitary matrices
Complex conjugate:
Adjoint:
Hermitian matrices:
Unitary matrices:
Inner product:
*A** ~
)( AAA T
AA
1 UU
Self-adjoint. Symmetric matrices in real space.
Orthogonal matrices in real space.
yxyx yx,The inner product of vectors x and y is
yxyx
yxyx
xyyx
AA
yAxyxAAA lhs))(rhs :(Proof
*
Unitary transformation: A unitary transformation preserves the inner product of complex vectors:
Orthogonality conditions:
yxyx UU
jkkiji
jkikij
uu
uu
*
*
Conjugate transpose. Sometimes A* in math books.
17
3.5 Diagonalization of matrices
Example: Diagonization of the moment of inertia.Angular momentum of a rigid body rotating around the origin. Let us consider one mass element m (I mean dm) inside the rigid body. The actual angular momentum takes the integration form.
IωL
ωrrωrωrvrL
or ,
)()(
22
22
22
22
2
z
y
x
z
y
x
jjiijjjiii
zrzyzx
yzyryx
xzxyxr
m
L
L
L
xxrmxxrmL
rmmm
We can rotate the coordinates so that the moment of inertia matrix I is diagonalized in the new coordinate system. If the angular velocity is along a principle axis, the angular momentum will be in the same direction as the angular velocity .
ors)eigen vect es,(Eigenvalu sum) (no '
then,tors)column vec are ( ),,(~
Let
~~
'00
0'0
00'~
321
3
2
1
iii I
I
I
I
vIv
vvvvR
I'RRIRRII'
18
Eigenvalues and eigenvectors:
For an operator A, if there is a nonzero vectors x and a scalar such that then x is called an eigenvector of A, and is called the corresponding eigenvalue.
A only changes the “length” of its eigenvector x by a factor of eigenvalue , without affecting its “direction”.
,xAx
0)( xxx AA
For nontrivial solutions, we needThis is called the secular equation, or characteristic equation.
.0)det( A
Example: Calculate the eigenvalues and eigenvectors of .52
34
A
1
100
52
34
2
30320
52
34
.7,2052
34)det(
22
2
11
1
21
v
v
A
yxy
x
yxy
x
19
Eigenvalues and eigenvectors of Hermitian matrices:1) The eigenvalues are real.
2) The eigenvectors associated with different eigenvalues are orthogonal.
Physicists like them.
Proof:
.0 then If
. then If
0)(
*
*
*
jiji
ji
ijijij
jiji
jijijj
ijijii
ii
ji
j
jj
ii
AAAAA
AA
AA
20
Read: Chapter 3: 4-5Homework: 3.4.4,3.4.5,3.4.7,3.4.8,3.5.6,3.5.8,3.5.9,3.5.12,3.5.27Due: September 30
21
September 28 Normal matrices
3.6 Normal matrices
Normal matrices: 0],[ AAAAAA
1) A and A+ have the same eigenvector, but with conjugated eigenvalues.Proof:
2) The eigenvectors of a normal matrix are orthogonal.
Proof:
.0
00][
00
.0][0],[ then ,Let . Suppose
*
AB
BBBB
BBBB,
BBBBB
BB,AAABA -λ
.0 then If
0*
ji
jijijijijiji
jiji
ji
jiii
j
AAA
A
22
More about normal matrices:1)Hermitian matrices and unitary matrixes are both normal matrices. However, it is not the case that all normal matrices are either unitary or Hermitian.2)A normal matrix is Hermitian (self-adjoint) if and only if its eigenvalues are real.A normal matrix is unitary if and only if its eigenvalues have unit magnitude.
3)Every normal matrix can be converted to a diagonal matrix by a unitary transform, and every matrix which can be made diagonal by a unitary transform is normal. Proof:
(unitary) 1
)(Hermitian
0],[
1
*
*
AA
AA
A
AAA
.0],[0],[ then diagonal, is andunitary is where, if Also
.
00
thentors,column vec as rseigenvecto usingmatrix unitary a ),,,,(Let
. and , that so buildcan we,0],[ If
1
21
AAΛΛΛUAUUΛ
AUUΛ
U
AAA
n
n
ijjiiiii
23
24
Reading: Spectral decomposition theorem:For any normal matrix A, there exists a unitary matrix U so that where is a diagonal matrix consists of the eigenvalues of A, and the columns of U are the eigenvectors of A.
, UΛUA
kkk
k
nn
n
2
1
2
1
21 ),,,( UΛUA
More explicit form:
Apply to functions of matrices: .kkk
kff A
kkm k
km k
kkmkm
m
kkkkm
m
mm
m
mm
faaaf
xaxf
AA
)(
:Proof
.1 Especially k
kk
25
Read: Chapter 3: 6Homework: 3.6.3,3.6.4,3.6.6,3.6.10,3.6.11Due: October 7