chapter 3 linear algebra mathematical methods in the physical sciences 3rd edition mary l. boas...

Chapter 3 Linear Algebra

Mathematical methods in the physical sciences 3rd edition Mary L. Boas

Lecture 7 Matrix

1 Introduction

- Algebra & Geometry

- Vector

- Change of coordinates (transformation)

H. W. (Due Apr. 16th)

Chapter 3

3-2. 8, 9

3-3. 1

3-5. 12, 16, 21, 37

3-6. 6, 15

3-7. 23, 24

“The problems similar to the above ones will appear in the midterm exam.”

2. Matrix; row reduction ( 행렬 ; 행줄이기 )

- A matrix is just a rectangular array of quantities, usually enclosed in large parentheses.

.6,0,3,2,5,1

number)(column number), (row:

matrix 2)by 2(32:603

232221131211

AAAAAA

- Transpose of a matrix

jiij A

A matrix23:

- Sets of Linear Equations

.3,2,1,

- Augmented matrix

(a) Eliminate the x terms in the other two equations by using the first equation.

ex. 2) – 1) x 3 2) , 3) – 1) 3)

(b) For convenience, interchange the second and third equations

(c) Eliminate the y terms by using the second equation.

ex. 3) – 2) x 5 3)

111100

(d) Eliminate the z terms by using the third equation.

ex. 1) + 3) / 11 1) , 2) – 3) / 11 2)

111100

(e) finalizing

2/3001

- Allowed rules

i. Interchange two rows

ii. Multiply (or divide) a row by a (nonzero) constant

iii. Add a multiple of one row to another; this includes subtracting, that is,

using a negative multiple.

5411row 2nd with therow1st with the

We can not get an answer. The equations are inconsistent.

- Rank of a Matrix

- The number of nonzero rows remaining when a matrix has been row reduced is called the rank of a matrix.

rank: 3 rank: 2 ‘Inconsistent’

- For example 2, the rank of A is 3, but the rank of M is 2. In this case, the equations are inconsistent [(rank of M) < (rank of A)].

a. If (rank M) < (rank A), the equations are inconsistent and there is no solution.

b. If (rank M) = (rank A) = n (number of unknowns), there is one solution.

c. If (rank M) = (rank A) < n , then R unknown can be found in terms of the

remaining n – R unknowns.

3. Determinants; Cramer’s rule ( 행렬식 ; Cramer 의 규칙 )

- We have said that a matrix is simply a display of a set of numbers; it does not

have numerical value. For a square matrix, however, there is a useful number

called the determinant of the matrix.

- Evaluating determinants

.det, bcaddc

i) 2 by 2

3333231

2232221

1131211

- When removing the row and the column containing the element a_ij, we have the remaining determinant, M_ij, called a minor of a_ij.

ii) nth order matrix

3333231

2232221

1131211

(n-1) by (n-1) determinant

- Finally, multiply each element of one row (or one column) by its cofactor and add the results.

1detjori

ijijji MaA

- Sign

- cofactor : ijji M 1

“Laplace’s development”

.1483851141237

.14821351112

i) method 1 (third column)

ii) method 2 (first row)

- Useful facts about determinants

1. If each element of one row (or one column) of a determinant is multiplied by a

number k, the value of the determinant is multiplied by k.

2. The value of a determinant is zero if,

(a) all elements of one row are zero

(b) two rows ( or two columns) are identical

(c) two rows (or two columns) are proportional.

3. If two rows ( or two columns) of a determinant are interchanged, the value of

the determinant changes sign.

4. The value of a determinant is unchanged if

(a) rows are written as columns are columns as rows

(b) we add to each element of one row, k times the corresponding element of

another row, where k is any number (and a similar statement for columns).

Example 2. Find the equation of a plane through the three given points (0,0,0),

(1,2,5), and (2,-1,0)

Example 4. Evaluate the determinant

1. Subtract 4 times the fourth column from the first column, and subtract 2 times the

fourth column from the third column

2. Do a Laplace development using the third row

3. Add the second row to the third row.

4. Do a Laplace development using the third row.

- Cramer’s rule

- We can use this method when you get the solution for the n linear equations

(in case that D 0).

- Denominator: determinant of the matrix with coefficients in the left side (M)- Numerator: for x, replace x part in M with right side part, and take the determinant.

for y, replace y part in M with right side part, and take the determinant.

- Rank of a Matrix

To find the rank of a matrix, we look at all the square submatrices and find their

determinants. The order of the largest nonzero determinant is the rank of the

matrix.

cf. submatrix: a matrix remaining if we remove some rows and/or columns.Example 6.

The submatrices of the original matrix are four (123, 124, 134, and 234).

Because the first and the second columns are absolutely the same, the determinants

of 123 and 123 should be zero. The absolute values of 134 and 234 should be the

same. For this reason, we need to check only 134.

3211submatrix

“The rank should be less than 3.”

Lecture 8 Vector

4. Vector ( 벡터 )

- Notation

yx AAA ,

- Magnitude 222zyx AAAA A

- Addition of vectors

addtionfor law eassociativ:

additionfor law ecommutativ:

CBACBA

- Multiplication by a constant & subtraction

- Unit vector A

- Vectors in terms of components

zzyyxx

BABABA

- Multiplication of vectors 1: scalar product

zzyyxx BABABA

CABACBA

on of projectiontimes

- Angles between two vectors using scalar product

example) Find the angles between these two vectors

1,3,2,9,6,3 BA

14,143963

21193623

- Perpendicular and Parallel vectors

- Multiplication of vectors 2 : vector product

BABABABA

productvector :

kijjikikjkji

kkjjii

zyxzyxzyx

AAABBBAAA

kjikjiBA

example 4. 2,3,1,1,1,2 BA

112 kji

5. Lines and planes ( 직선과 평면 )

- A great deal of analytic geometry can be simplified by the use of vector notation.

Such things as equations of lines and planes, and distances between points or

between lines and planes often occur in physics, and it is very useful to be able to

find them quickly. Vector notation will help you do these more easily.

‘points vectors’

To determine a specific line, we need one point and a slope (= two points).

zzyyxx

,, Slope cf. 121212

- Straight Lines

,, Variable

,,pointGiven 1110

)0,, line,straight a ofequation symmetric(,

line)straight a of equations c(parametri

tort ArrArr

To determine a specific plane, we need a point and a normal vector.

plane) a of (equation ,0

dczbyaxorzzcyybxxa

- Planes

Example 1. Find the equation of the plane through the three points A(-1,1,1),

B(2,3,0), C(0,1,-2).

- Because points are given, what we have to do is to find the normal vector.

0023826 :plane theofEquation

3,0,1),1,2,3()0,1,1(0,3,2

Example 1. Find the equation of the plane through the three points A(-1,1,1),

B(2,3,0), C(0,1,-2).

Example 2 Find the equation of a line through (1,0,-2) and perpendicular to the

plane of Example 1

1 : line theofequation

Example 3 Find the distance from the point P(1,-2,3) to the plane 3x-2y+z+1=0.(Use the dot product.)

RQ PR: distance we want to know

Q : any point on the plane

product.)dot the torelated is projection (The cosPQPR

- Choose Q in the easiest way, e.g., Q=(0,0.-1) or (1,2,0) (as in the text)

.14/1114/1,2,34,2,1

14,1,2,3,4,2,13,2,11,0,0

.for ,

NNnnNN

PQPQPR

Example 4 Find the distance from P(1,2,-1) to the line joining P1(0,0,0) and

P2(-1,0,2). (Use the cross product.)

./ where,sin AAaa PQPQPR

.5215/1,2,1)2,0,1(

1,2,1,2,0,1

PPPQPP

Example 5 Find the distance between the lines, r=i-2j+(i-k)t, r=2j-k+(j-i)t.

n nBABAn PQ/

321,1,1,1,4,1

0,1,1,1,2,0

1,0,1),0,2,1(

Example 6. Find the direction of the lines of intersection of the planes

Note) the intersection lines are perpendicular to both the planes.

Example 7. Find the cosine of the angle between two planes

Note) The angle between the planes is the same as the angle between the normal vectors to the planes

Lecture 9 Matrix operation

6. Matrix operations ( 행렬계산 )

- Matrix equations

- Multiplication of a matrix by a number

.1,5,7,1,3,2

ivuisryx

kfkdkb

kekcka

T 32or3

232 AAjiA

- Addition of Matrices

Note) Addition (or subtraction) can be done with the same type of matrices

231cf.

217734

421321

- Multiplication of matrices

The element in row i and column j of the product matrix AB is equal to row

i of A times column j of B. [(# of row of A) = (# of column j)]

In index notation,

kjikij BAAB

dhbfdgbe

chafcgae

Example 1

413371532113

423472542214

Example 2. Find AB and BA

“not commutative”

- Zero matrix

: zero or null matrix means one with all its elements equal to zero.

42 2MMcf.

- Identity matrix or Unit matrix

AAIIAI

- Operation with Determinants

BABAAB detdetdetdet

- Applications of matrix multiplication

kMrkMr 1Then,

2 Method

321 Method

- Inverse of a Matrix

IMMMM 1 1

ijijT MmC 1, ofcofactor where

Example 3. 3det,

10 : row 3rd

10 : row 2nd

03 : rowst 1

3461- TC

Finding the cofactor

- Rotational matrices

cossin

sincos

cossin

sincos

cossin

sincos

Lecture 10 Linear transformation

7. Linear combinations, linear functions, linear operators ( 선형결합 , 선형함수 , 선형연산자 )

- A function of a vector, say f(r), is called linear if

rrrrrr afaffff and,2121

21212121

),1,3,2( 1 Ex.

rrrrrrrr

rrArAr

rrrArArrArr

afaaaf

linear

not linear

- F(r) is a linear vector function if

rFrFrFrFrrF aa and,2121

- Linear operator

AABABA kOkOOOO and

- Matrix operators, Linear transformations

Moving a point to some other point

mapping or transformation

M : transformation matrix (linear operator)

ii) two sets of coordinates (x,y) (x’,y’) & one vector r = r’

i) One set coordinate & r R

- Orthogonal transformation

: linear transformation preserves the length of a vector.

:maxtrix Orthogonal

:mation transforOrthogonal

cf. det M = 1 : rotation, det M = -1 : reflection

222222222222

dbcdab

cdabca

cdabdbca

yxydbxycdabxcadcxbaxyx

T MMMM

prove)

detdet1detdetdetdetdet 2

MMMMMIMM TTT

Example 3. Find what transformation correspond to each of these matrices

1BADABCBA

reflectiondetdetdet,1detdetdet,1det

rotation,1det

ABDBACB

rotation 12032

ii) B : y -y, reflection through the x axis.

iii) C

- We want to find the reflection line. The vector lying on the reflection line is not changed by the reflection.

3say,,3

Analyzing the D in the same way,

3say,,3

- Rotation in 2 Dimensions

cossin

sincos

cossin

sincos

vector rotated(active)

axes rotated(passive)

change of basis

kjikji ,,,,

- Rotations and Reflections in 3 Dimensions

0cossin

0sincos

0cossin

0sincos

cos0sin

sin0cos

rotating along the z-axis

rotating along the z-axis + reflection through xy plane

rotation along the y axis

8. Linear dependence and independence ( 선형종속과 선형독립 )

Three vectors A=i+j, B=i+k, C=2i+j+k are linearly dependent,

because A+B-C=0.

ly,Equivalent

- Linear independent of functions

0332211 xfkxfkxfkxfk nn

In this case, these functions are linearly dependent.

ex. 1)

ex. 2)

xx 22 cos1,sin

xx cos,sin

linearly dependent

linearly independent

t.independenlinearly are functions thethen,

wronskian

if and,1order of derivative have ,,, If

xfxfxf

xfxfxfxf

nxfxfxf

Example 1. xx sin,,1

Example 2. xxxx sin32,sin,

sin30sin0

cos32cos1

sin32sin

- Homogeneous equations (right sides are zero)

yx x=y=0, (rank 2) = (# of unknowns)

(trivial solution)

all points on x+y=0, (rank 1) < (# of unknowns)

(nontrivial solution)

Example 4. For what values of does the set of eqs. have nontrivial solutions?

5,0,042

9. Special matrices and formulas ( 특별한 행렬과 공식들 )

- Summary “Please take a look at (9.1) and (9.2)

- Index notation

kjikij BAAB

- Kronecker

jiifij

dxmxnx

nmifdxmxnx

coscos

,coscos

- More useful theorems

ABCDABCD

ABCDABCD TTTTT

Lecture 11 Diagonalizing matrices

10. Linear vector space ( 선형 벡터공간 )

(Please read this section individually.)

Moving a point to some other point

mapping, transformation, or deformation

M : transformation matrix (linear operator)

- One set coordinate & r R

11. Eigenvalues and eigenvectors; diagonalizing matrices ( 고유값과 고유벡터 ; 행렬의 대각화 )

Some vectors are not changed in direction by transformation.

const. where rRMrR

r : eigenvectors (characteristic vector): eigenvalues

- Definition of eigenvalue and eigenvector

- Eigenvalues ( 고유값 )

.0)2(2

,02)5(or

:condition rEigenvecto22

μxyx-

seigenvalue.6or 1

,0674)2)(5(

.matrix ofequation sticcharacteri 022

,0 other thansolution afor Condition

According to the definition,

- Eigenvectors ( 고유벡터 )

- From the above results,

6.for 02

1for 02

25 through

02on 6

Any points in two straight lines can be eigenvectors.

6for 1

4) Diagonalizing a Matrix ( 행렬 대각화 )

6.for 622

625 1,for

.difference no :60

06 cf.

tors)(unit vec

where,

CDMCyy

Representing with the matrix operations,

DCDCMCCCDMC

DMCC 1

- Matrix D has elements different from zero only down the main diagonal,

diagonal matrix.

- D is called similar to M.

- When we obtain D given M, we have diagonalized M by a similarity

transformation.

5) Meaning of C and D (C 와 D 의 의미 )

(x’, y’) rotated through from (x, y)

).','(' and ),( where'

.lyspecifical cossin

sincos where'

cos'sin'

sin'cos'

1 rDrMCCRrMCRCMrR

YXRYXRCRR

- D=C-1MC is the matrix which describe in the (x’, y’) system the same

deformation (or transformation) that M describes in the (x, y) system.

- If C is chosen to make D=C-1MC diagonal, the new axes are along the

directions of the eigenvectors of M.

- The matrix C which diagonalizes M is the rotation matrix when the (x’, y’)

axes are along the directions of the eigenvectors of M.

12. Applications of diagonalization ( 대각화의 응용 )

- Central conic section (ellipse or hyperbola) with center at the origin

KByHxyAx

''''or cossin

sincos''

cossin

sincos

xMCCyxK

xMCCCCyxK

CyxCyxyxyxyx

If C is the matrix which diagonalizes M, the above is the equation of the conic relative to its principal axes.

Example 1. 30245 22 yxyx

.arccos cf. .30'6''

section) previous (from .6 ,1 seigenvalue ,22

2530245

51matrixrotation

xyxyxyx

This idea can be applied to three (or more) dimensions.

Example 2. 24226 222 zyzyxyx

.3,4,1

)3)(4)(1(1213

matrix thisofequation sticCharacteri

.24'3'4'or 24

''' 222

equation, above the to3,4,1 Applying

Let’s find C.

.3 when ,, ;4 when ,, ;1 when ,0,

Example 3. Find the characteristic vibration frequencies

).222()(

xyyxkyxkkykxV

3,1 ,0313421

s,eigenvalue thefind To

(continued)

,in matrix thechange variable themake To

,in variablenew a find To

UCCUCCy

Lagrange’s equation

tBytAx

kyymkxxm

yxkyxmVTL

.conditions initial depending sin' ,sin'

'.3' ,''

222122

(continued)

Finding the orthogonal transformation matrix C,

3for 2

1 ,1for

3for 3 21

12 ,1for 1

yxyyxxy

(continued)

phase ofout ).)(( .sin22

' .0' ,0For

phasein ).)(( .sin22

' .0' ,0For

,sin ,sin

tBytAx

‘Characteristic (or normal) modes of vibration’‘Characteristic (or normal) frequencies of the system’

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