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