data-think linear

Matricesand vectors, which underlielinearollf'lI'£lI1~^...."ll(Chaps. 7 and allow us torA-n,rA(1Ant-

numbers or functions in an ordered and form. Matrices can hold enormous amounts of data-think of a network of millions of connections or cell phone connections-

in a form that can be The main of 7 is how

to solve systems linear linear

fI.,..L ...L..LU..L'U'..L~L..L..L... fI.,..L'U' .. ..LU, and vector spaces are related. "'-'Ju......,"',...,.IL

'U'ILJ'..L"-'.A...LA.U. Linear is an active field that has many u.IJltJ..L..L..."'..L'U'..L.h.J

" - ' v ...)'.l..l. ...'.l..l..l..1.vL.J~ and others .

...JL.JL'~IIJ...lLLJ 9 and 10 extend calculus to vector calculus. We start with vectors from linear

and vector differential calculus. We differentiate functions of several variables and discuss vector differential such as and curl. 10

extends to over curves, and

obtammz new of theorems and Stokes allow

us to transform these into one another.

can be found

can be studiedIlIlIlFrIV""II,lIlIl

moenenuentof the other in Part E on numerics.

7 or8 because

線性代數

向量微積分

梯度,散度,旋度

各章的學習 重點

in that it has many UftJ!fJ.1. ..U""UL.1.V.L.1.0 I'A1nrlr'l,11 ....t:llrU... A....,AjL ...~ 'lV'V ...J.L.L ...,.1..1..LJI.'VlJ, and other areas. It also contributes to a of mathematics itself.

..L ...~^....""AA....,_U" which are arrays of numbers or and are the

main tools of linear Matrices are because let us express amounts of data and functions in an and concise form. since matrices are we denote them letters and calculate with them All these features have made matrices and vectors very for pV1nrp'C'C'1no-

scientific and mathematical ideas.

The mix between Markov

processes, traffic and 7 is structured as follows: Sections 7.1 and 7.2 an intuitive introduction to matrices and vectors and their A-.n.ar.r,1-lIr,1l"'\C\

1 n l ' I l l r l 1 n c rmatrix The next block of that Sees. 7.3-7.51n.1I"'n,,,TlIrta

the most method for the Gauss

elimination method. This method is a cornerstone of linear and the method itself and variants of it appear in different areas of mathematics and in manyuftJ.J.1.JI. ...ULJlLV.L.L0.

It leads to a consideration of the behavior of solutions and such as rank of a

linear and bases . We shift to a that has

declined in in Sees. 7.6 and 7.7. Section 7.8 covers inverses of matrices.

The ends with vector spaces, inner spaces, linear and

I'A1nrlr'l,AC'1It1r~n of linear transformations. follow in 8.

rrereauisite: None.

Sections that may be omitted in a short course: 7.9.

ffp·fPypn,"p\, and Answers to Problems: 1 Part and

線性代數：矩陣，向量，行列式

線性系統(指聯立線性方程式)

第七章的學習內容 Matlab軟體處理矩 陣運算的功能很強 大，值得學習

SEC. 7.1 ""~,Trl.~oC' Vectors: Addition and Scalar I\Jl"II+II,",II'~"'+I""''''''

The basic concepts and rules of matrix and vector algebra are introduced in Secs. 7.1 and 7.2 and are followed linear of linear a main<:lnl'"'\llf"'<:lt"1IArII

in Sec. 7.3.

Let us first take a look at matrices before we formalize our discussion. Amatrix is array of numbers or functions which we will enclose in brackets. Forp V ' : l r r l - n ! p

[

O~3

^-0.21

~:l

^alla21a31 ^a22a32^al2

[:::

^4x

^2x2l

^a2

[:J

Consider the data in

and

are matrices. The numbers are calledentriesor, less elements of the matrix. The first matrix in has tworows,which are the horizontal lines of entries .

... "JL""",",,""""""""'L''''''''"'',it has threecolumns, which are the vertical lines of entries. The second and

1i"1r"a'::BJf"lr'1i~tl.llC' which means that each has as many rows as columns-

3 and 2, The entries of the second matrix have two their

location within the matrix. The first index is the number of the row and the second is the

number of the so that the is identified. For

a two is in Row 2 and Column 3, etc. The notation is standard

.... .lL ''''"''u, In(~lU~jlnlgthose that are not square.

a row or column are calledvectors. the fourth matrix in has one row and is called a vector. The last matrix in has one

column and is called a Because the of the of entries was

to the of an element within a one index suffices for

vectors, whether are row or column vectors. the third of the row vector in is denoted

We are given a system of linear equations, briefly a system,such as 4xI +6x2 +9x3 = 6

6XI - 2X3= 20

where^{Xl,X2, X3}are theunknowns.We form thecoefficient matrix,call itA,by listing the coefficients of the unknowns in the position in which they appear in the linear equations. In the second equation, there is no unknownX2, which means that the coefficient ofX2is

°

and hence in matrix a22 = 0, Thus,

第四章4.0節已講過相同的觀念

元素 元素

Row 列

Column 行

方陣(行數=列數)

第2列第1行 (雙註符號) (元素位置)

行向量 列向量

必須先按照順序排 好，缺項補為0 變成AX=C矩陣形式

(一定要會的技巧！)

CHAP. 7 Linear Algebra: Matrices, Vectors, Determinants. Linear Systems

A=[:

6

-:J ^{A= [:}

6 9

2~J

0 We form another matrix 0 -2

-8 -8 10

by augmenting A with the right sides of the linear system and call it the augmented matrix of the system.

Since we can go back and recapture the system of linear equations directly from the augmented matrixA, A

contains all the information of the system and can thus be used to solve the linear system. This means that we can just use the augmented matrix to do the calculations needed to solve the system. We shall explain this in detail in Sec. 7.3. Meanwhile you may verify by substitution that the solution is^Xl =3,^X2 =

!,

^X3 = -l.

The notation^{Xl, X2, X3}for the unknowns is practical but not essential; we could choose^X,y,Zor some other letters.

in

Sales figures for three products I, II, III in a store on Monday (Mon), Tuesday (Tues), . '. may for each week be arranged in a matrix

Mon Tues Wed Thur Fri Sat Sun

[40

^{33 81 0 21 47} ₃₃

J

A= 0 12 78 50 50 96 90 ^.I1

10 0 0 27 43 78 56 III

If the company has 10 stores, we can set up 10 such matrices, one for each store. Then, by adding corresponding entries of these matrices, we can get a matrix showing the total sales of each product on each day. Can you think of other data which can be stored in matrix form? For instance, in transportation or storage problems? Or in listing distances in a network of roads?

have discussed. We shall denote matrices

the in thus A == and so

on. an x n matrix m n we mean a matrix with m rows and n

columns-rows come first! m X n is called the of the matrix. Thus an m X n matrix is of the form

The matrices in are of sizes 2 X 3, 3 X 3, 2 X 2, 1 X 3, and 2 X 1,roC'·nor-i-lI'{Tol'{T

Each in (2) has two The first is the row number and the second is the column number. Thusa21 is the in Row 2 and Column 1.

If m == n, we call A an n X nsquare Then its .....l.u.~V~I....L"..l. contammzthe entries

all,a22, . " , annis called the of A. Thus the main of the two square matrices in areall,a22, a33and

matrices are"1n>n·r1""f' ln-rI-.:T lI"Y"Y\"1n>,n.-r1"n"Y\1" as we shall see. A matrix of any size m X n is called a rectanzurar.u , "',this includes square matrices as a case.

係數矩陣 擴大(增廣)矩陣

矩陣形式之銷售圖表

日期 三種商品

主對角 方陣(行數=列數)

行數≠列數 粗體且大寫

注意下標的表示法與代表意義

方陣A中定義軌跡(trace):

trace A=a11+a22+...+ann

SEC. 7.1 Vectors: Addition and ScalarJY'''''U'''lIl-'lI'~U'''II_11

A vectoris a matrix with only one row or column. Its entries are called the comnonents of the vector. We shall denote vectors by lowercase boldface letters a, b,'" or by its general component in brackets, a == [aj

J,

and so on. Our special vectors in (1)

that a (general) row vector is of the form

a == [al a2 an]. instance, a == [-2 5 0.8 0 1].

A column vectoris of the form bl

4

b2 For 0

-7 b_m

What makes matrices and vectors useful and"t"'\l1lll~i"lIr>llll)lrh:T

the fact that we can calculate with them almost as

now introduce rules for addition and for scalar-rY\rlli"1I1nI1l/"'l1li"-'A11I \~~~"""..L"".~I-'..L.~v^...""..L'J~~

that were suaaested

follows first need the""f'"".""",,~e-o.'I-

Two matricesA == [ajk ] and == [bj k ] are have the same size and the.nr..1I·....ac1~r...rArI-.rAnr aI2 == b_{12 ,}and so on. Matrices that are not of different sizes are different.

if

Let

and _B=

[ 4 0]

3 -1 Then

A=B if and only if

all = 4, al2= 0,

a21 = 3, a22= - 1.

The following matrices are all different. Explain!

粗體且小寫

在兩矩陣的對應位 置，每個元素都應 該相等

相等性

2x2 2x2 兩個矩陣大小(nxm)相等

分量

CHAP. 7 Linear Matrices, Vectors, Determinants. Linear ""''1.;J ... " , ....

The of two matrices A == [ajk] and == [bj k ] the same size is written A

+

and has the entries ajk

+

bj kobtained the entries of A and Matrices of different sizes cannot be added.

As a case, the

+

of two row vectors or two column vectors, which must have the same number of '-''U'J_l-J.IJ'U'.Lll~''''''J.lllI.-l.J. is obtained the I"r"I,rl"'t=AC'1'"'\I,,;n,'iln 0-

If

0] __ [3 1 52 3

0' then A+

2],

in Example 3 and our present cannot be added. If a= [5 7 2] and

a+ 9

An application of matrix addition was suggested in Example 2. Many others will follow.

The of anym X nmatrix A == [ajk] and anyscalar^C\ L L ...LJCL"-''''"'..L

cA and is them X n matrix cA == obtained c.

Here (- is written - A and is called the of A...JCL~_JCJCJC. . .,-'LJC

written -kA. A

+

and is called the.riit~t-tl>1r'.tOn/"bo

must have the same

2 then

[

2,7 -1.8J

If A= 0 0.9, then

9.0 -4.5

[

-2.7

=

°

-9.0

1.8J [ 3

-0.9, 19° A= °

4.5 10

-2J

^{1 , OA=}

[0 OJ

^{0 0.}

-5

° °

Ifa matrix shows the distances between some cities in miles, 1.609B gives these distances in kilometers.

Matrix Scalar From the familiar laws for the

addition of numbers we obtain similar laws for the addition of matrices of the same size m Xn,

A+ +

sizem X that them X n matrix with all entries this is a vector, called a

相加(減)運算(兩矩陣有相同nxm形式，對應位置元素進行相加(減))

2x3 2x3

純量相乘(矩陣中全部元素都必須乘上c值) 矩陣

(c值可乘於任何一列或任何一行) 行列式

交換律 結合律

零矩陣(所有的元素都是0)

行列式 矩陣

請注意：

If A is a matrix, then det (cA) = |cA| = c ⁿ det (A) = c ⁿ |A|

行列式與純量相乘時，純量可乘入任一行或 任一列

SEC. 7.1 An ....,..,. ... ".. Vectors: Addition and ScalarI\JlIII1'"III""\I,,-~1'"I^...n

Hence matrix addition is commutative and associative [by Similarly, for scalar multiplication we obtain the rules

and (3b)].

+4D, 8

+C+

-4 w= -10

inurcatmz which of the reasons

+

+ O· E, E- + v) + 2w, 8(0 + v) +

+ (v - w), C + Ow,

2B + OA+ 0.2B - 2.4A

5A +0.25B, 5A +0.25B+ C 8B - 2B,

+ 0.4C - 0.4D,

2 0

E= -4 3

-3

1.2 2

0 v= -1

-2.5 3

Find the rules in(3)or(4) are not

of Ain the five matrices in

nUUi1.ill.lV. Give reasons

I-4v,n1"V\""la3 are all different.

Double notation. If you write the matrix in

-L-i1""'....,LL.LIJ.L'V2in the form A= [ajk],what is

3. Sizes. What sizes do the matrices inrsxamutes 1, 2, 3, and 5

Main dI3,2011al. What is the main

HV'r:lrnnl~ I? Of A and B inHV'CllYlnip

5.. Scalar in 2 shows the

number of items what is B of units sold if a unit consists of 5 items and (b) 10items?

6. If a 12X 12 matrix A shows the distances between 12cities in kilometers, how can you obtain from A the

matrix B these in miles?

7. Addition of vectors. add: A row and a column vector with different numbers of compo- nents? With the same number of Two with the same number of components numbers of zeros? A vector and a A vector with four components and a 2 X 2 matrix?

Let 16.. 15v - 3w - Ou, 15v - 3w, +3C,

4.5w - 1.2u +O.2v

-2 5 5 2 0

of forces. If the above vectors v, w

A= 4 4 8 -5 3 -4 represent forces in space, their sum is called their

resultant.Calculate it.

-3 0 -4 2 -4

definition, forces arein equilibrium

6 -2 -3 if their resultant is the zero vector. Find a force p such

2 -4 2 0 that the above u,v,w,and are in equilibrium.

19.. rules, Prove (3) and (4) for general 2X 3

0 -1 -1 2 matrices and scalars c and k.

CHAP. 7 Linear Matrices, Vectors, Determinants. Linear

where

Mesh Incidence Matrix. A network can also be themesh incidence matrixM= [mjkJ,

and a mesh is a loop with no branch in its interior (or in its Here, the meshes are numbered and directed in an fashion. Show that for the network in 157, the matrix has the form, where Row to mesh 1, etc.

o

+1 if branch kis in mesh [ ] ] and has the same orientation mjk = -1 if branch kis in mesh

[Z]

and has the orientation

oif branch kis not in mesh

[JJ

o -1 -1

Sketch the three networks corresponding to the nodal incidence matrices

I-~

^{-10 00 00}

I-~

^{-10 -10 -10}

~l

L 0 ^{L ~ -...J}

0 0 0

-1 0

o .

3

20. TEAM PROJECT. Matrices for Networks. Matrices have various engineering applications, as we shall see.

For instance,they can be used to characterizeconnections in electrical networks, in nets of roads, in production processes, etc., as follows.

(a) Nodal Incidence Matrix. The network in 155 consists of sixbranches(connections) and fournodes (points where two or more branches come together).

One node is thereference node(grounded node, whose voltage is zero). We number the other nodes and number and direct the branches. This we do arbitrarily.

The network can now be described a matrix

=[ajkJ,where

1 if branchkleaves node

0

ajk = 1 if branchkenters node

0

oif branchkdoes not touch node

0.

A is called thenodal incidence matrixof the network.

that for the network in 155 the matrix A has form.

1 1 0 -1 ⁰

0 0 0 -1

M= 0 -1 1 0

1 0 1 0 0

Network and matrix in Team

""::""

Branch 1 2 3 4 5 6

Node

l~

^{-1 -1 0 0}

^{-~ J}

Node 1 0 1 1

Node 0 1 0 -1

Network and nodal incidence matrix in Team

Find the nodal incidence matrices of the networks in Fig. 156.

Electrical networks in Team

SEC. 7.2 Matrix Multiplication

Matrix multiplication means that one matrices matrices. Its definition is standard but it looks artificial. Thus you have to study matrix multiplication carefully, multiply a few matrices together for practice until you can understand how to do it. Here then is the definition. follows

The C== AB (in this matrix == [b_{j k ]} is defined if and C == ^{[Cjk ]} with entries

of anm X nmatrix A == [ajk] times an r Xp if r == n and is then the m X p matrix

The condition r == nmeans that the second must have as many rows as the first factor has n. A of sizes that shows when matrix -n-1Il-.1-t11nl1''''nl-t-wr.n

is is as follows:

A

Xn][nXp]

C [mXp].

the amutunucauon

The is obtained in the row of

in the kth column of and then thesenP~W:-{;~GIUll~t"l.S.For instance,

+

a22b21

+ ... +

and so on. One calls this into columns. Forn == 3, this is illustrated

Notations in a

...IL'to..&.....J..J..J.IIJ.I."'-"CI. Note that after where we shaded the entries that contribute to the calculation of discussed.

Matrix will be motivated its use in linear transtormations in this section and more in Sec. 7.9.

Let us illustrate the main of matrix -rYlll"11i-1I1nl1'-'rl't-Ir.n

matrix multiplication also includes a matrix a vector is a special matrix.

AB

-2

o

-4

1] [22

8 = 26

1 -9

-2

-16 4

43

-37

Hereell= 3 . 2+5 . 5+(-1) . 9= 22, and so on. The entry in the box isC23 = 4 .3+0 . 7 +2 . 1= 14.

The product BA is not defined.

矩陣相乘=矩陣×矩陣

=Aj × B^k( 即(A的第j列向量) × (B的第k行向量) )

前面矩陣的行數=後面矩陣的列數 (如此才能進行one by one相乘，

而後再相加)

小心！容易計算錯誤

以實際例子說 明較容易了解 一定要成立的條件！

c21=A的第2列 × B的第1行

=a

21

b

11

+a

22

b

21

+a

23

b

31 n=3

=Σ a

l =1 2 l

b

l1

CHAP. 7 Linear Matrices, Vectors, Determinants. Linear" J V _ : : I L . . . ...::0

[ 4 2] [3]

⁼

[4 .3

⁺

2.5]

⁼

[22]

1 8 5 1 . 3+ 8 . 5 43

whereas is undefined.

This is illustrated by Examples 1 and 2, where one of the two products is not even defined, and by Example 3, where the two products have different sizes. But it also holds for square matrices. For instance,

[ 1 1] [-1 1] [0 0]

100 100 1 - 1 = 0 0 but

[ -1 ][1 1] [99

1 - 1 100 100 = -

99 99]

-99

Itis interesting that this also shows that = 0 doesnot necessarily imply BA= or = 0 or = O. We shall discuss this further in Sec. 7.8, along with reasons when this happens.

examntcs show that in matrix-rl>1I"'r"rl1I1I.01"Clthe order must be observed Otherwise matrix satisfies rules similar to those fornll"1111r"\ h,01l"'Cl

written kAB orAkB written ABC

== AC

+

== CA

+

and C are such that theOV1t"-roC'C'lI,,,,nC' on the left are kis any scalar. is called the associative and are called the rli,~t-riihl1Ihi'T{;}'laws.

Since matrix is a of rows into we can write the

rlo1rll-n11-n.nr- formula more as

j == ...,m; k == ... ,p,

where aj is the azreement with

row vector of and is the kth column vector of so that in

+ + ... +

無法計算！

矩陣相乘的特殊性質

矩陣相乘不滿足交換律

矩陣相乘時，矩陣 的順序不可以改變 (性質) AC=AD does not necessarily imply C=D (even A≠0)

A矩陣第j列向量 B矩陣第k行向量

前面矩陣的行數=後面矩陣的列數 (如此才能進行one by one相乘，

而後再相加) 有些版本用上標代

表行向量

SEC. 7.2 Matrix'VIY''-'I-'U'"''liA'-'''''''

IfA= [ajkJis of size 3X 3 and B= [bjkJis of size 3X4, then

(4) AB=

Taking al= [3 5

-1

^{J, a2= [4 0} 2J, etc., verify (4) for the product in Example 1.

are then assianec to different processors

which the columns of the 1I""\I ...., r " r I I r••n'l-

obtain

from (5), calculate the columns

o

4

7] [11

6 = -17

4 8

34]

-23

of AB and then write them as a single matrix, as shown in the first formula on the right.

Let us now motivate the "unnatural" matrix its use in transtormattons, Forn ^:=: 2 variables these transformations are of the form

and suffice to the idea.

instance, (6*) may relate an

plane. In vectorial form we can write (6*) as

will be discussed in Sec. 7.9.) For

to a in the

(6) y = [::] = Ax =

:::] [::]

B矩陣的行向量 矩陣相乘可提供電

腦平行運算，增快 運算速度

p次平行運算可同 時進行

矩陣相乘的動機：

要作線性轉換

轉換成矩陣 y1, y2系統與x1, x2

系統的關係

CHAP. 7 Linear Determinants. Linear ...^,/c...Oi""Y"1lC'

Now suppose further that the transformation, say,

<")-:"iV:"iICIII is related to a WIw2-system by another linear

(7) x= [::]

Then the is related to the we wish to express this relation

a linear too, say,

lAln·- ... \.f .... It-".111I indirectly via the xIx2-system, and

Substitution will show that direct relation is

y==

we obtain

+ +

YI ==

+ + +

Y2==

+ +

+ + +

with we see that

+ +

+ +

This proves that C == AB with the defined as in For matrix sizes idea and result are the same. the number of variables cnanzes, We then have m variables Y and n variables x andp variables w. The matrices and C == then

have sizes m X n, n X p, and m X p, And that C be the

"f"....n'rllIlI.ni"AB leads to formula in its form.This motivates matrix muuuiucauon.

transpose of a matrix its rows as columns its

columns as This also to thet"r<:llnCll""\r\('Aof vectors. a row vector becomes a column vector and vice versa. In for square we can "reflect"

the elements the main entries that are svmmetncauv

"f"r\(:"lI1"lI,n.nc.hrlwith to the main to obtain the HenceaI2 becomes

a21, a3Ibecomes a13,and so forth. 7 illustrates these ideas. Also note if A is the then we denote itst"r<"lnC1l1'"'\r\('O

If

-8

o

^then

x1, x2系統與w1, w2

系統的關係

代入

y1, y2系統與w1, w2

系統的關係

y= Ax = ABw=Cw，即C=AB (矩陣相乘 可應用於不同線性系統之間的轉換) (7)式代入(6)式中

x1 x2

c11 c12

c21 c22

x1 x2

整理成(8) 式做係數 亦即C=AB 比較

矩陣轉置

A → A

^T

(行、列位置互換)

矩陣乘法補充：齊

次座標系統的轉換

y

₁

, y

₂系統與 x₁

, x

₂系統的關係

1 11 1 12 2

2 21 1 22 2

y a x a x y a x a x

= +

= + →

^{1 11 12 1}

2 21 22 2

y x

y a a x

y a a x A

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ = ⎥ ⎢ ⎥ =

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

x

₁

, x

₂系統與 w₁

, w

₂系統的關係

1 11 1 12 2

2 21 1 22 2

x b w b w x b w b w

= +

= + →

^{1 11 12 1}

2 21 22 2

x w

x b b w

x b b w B

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ = ⎥ ⎢ ⎥ =

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

y

₁

, y

₂系統與 w₁

, w

₂系統的關係

1 11 12 11 12 1 11 12

2 21 22 21 22 2 21 22

y = = = w = w

x w

C

y a a b b w c c

A A

y B a a b b w c c

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

即

^{11 12 11 12 11 12}

21 22 21 22 21 22

C = =

c c a a b b

c c AB a a b b

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦

→

矩陣相乘可應用於不同線性系統之間的轉換

SEC. 7.2 Matrix ^{1\.11 - ...}

A little more compactly, we can write

[:

-8

o

Furthermore, the transpose [6 2 3JTof the row vector [6 2 3] is the column vector

[:l

^{= [6 2 3].}

...LU..L..Lk)J-1'Vk)v of an m X n matrix A == [ajk] is the n X m matrix A

that has first rowofA as its first the second row of as its second\...-U£'VH'''lr,,~and so on. Thus the ofA in is == [akj], written out

Asa case, ....~n.'nCl..-..~Cl-... n ...converts row vectors to column vectors and^{0A111l ...}rt:::..rC't::l>.I...T

us a choice in that we can work either with the matrix or its

L.1.UJL.l0I--fV0I\,.1qwhichever is more convenient.

Rules for are

Note that in the matrices are reversed the proofs as an exercise in Probs. 9 and 10.

We leave

Certain kinds of matrices will occur

most ones of them.

....rO.n111£:..nt"ll ...rin our and we now list the

~"lnnl1r1il1ptrj~ and ~Ke~T..Svmmetrrc Matrices, rise to two useful

classes of matrices. matrices are square matrices whose transpose equals the

行向量 列向量

A

^T

A → A

^T

(行、列位置互換)

研究所考過證明 (交大機械) (見補充資料)

｜A｜=｜A

^T

｜

特殊矩陣

對稱矩陣(R) 反對稱矩陣(S) 注意下標的差異

CHAP. 7 Linear Matrices, Vectors, Determinants. Linear "'\fC~T~lrnc:

matrix itself. matrices are square matrices whose the matrix. Both cases are defined in (11) and illustrated

[ 20 A= 120 200

120 10 150

200]

150 30

is symmetric, and

=[-; ~ ^=:]

is skew-symmetric.

For instance, if a company has three building supply centers C1,C2 ,C3 ,then A could show costs, say,^ajjfor handling 1000 bags of cement at center and^ajk(j =/=k)the cost of shipping 1000 bags from CjtoCi:Clearly,

ajk = akjif we assume shipping in the opposite direction will cost the same.

Symmetric matrices have several general properties which make them important. This will be seen as we proceed.

ipper trtangularmatricesare square matrices that can have nonzero diagonal, whereas any below the must be trtangutarll1nl':lIf"1"'U£ll.o~can have nonzero entries on and the

on the mainU-lU,F,VII..lU.lof a matrix may be zero not

-1 6

3 9 -3

a

9 2 3

lJUU!Onal Matrices.. These are square matrices that can have nonzero entries on

the main above or below the main must be zero.

If all the ^{rl1r:l1n A nr:II} entries of a matrix S are say, c. we call a because of any square matrix of the same size S has the same

effect as the a that

AS == SA == cA.

whose entries on the mainriI ...,~r..·rlI"'"^I are all 1, is called a

11"16:1I"tjt'7 ....c,.iI-,..,.." ...T\and is denoted or For formula becomes

== A.

注意特徵！

三角矩陣

上三角矩陣 下三角矩陣

對角矩陣

只有對角線 非全為0

純量矩陣 單位矩陣 S=c I

=c IA 交換律成立 A為任一矩陣，其必為一對

稱矩陣與反對稱矩陣的和

(見補充資料)

SEC.7.2 MatrixI\JII.I ...i"'lil,.."'ll ... i"'...

Supercomp Ltd produces two computer models PC1086 and PC1186. The matrix A shows the cost per computer (in thousands of dollars) and B the production figures for the year 2010 (in multiples of 10,000 units.) Find a matrix C that shows the shareholders the cost per quarter (in millions of dollars) for raw material, labor, and miscellaneous.

Quarter

PC1086 PCl186 2 3 4

[1.2 1.6]

Raw Components

B= [ : ^{8 6}

~]

^PC1086

A= 0.3 0.4 Labor

2 4 PCl186

0.5 0.6 Miscellaneous

1 2 3 4

[13.2

^{12.8 13.6}

15.6]

Raw Components

C=AB= 3.3 3.2 3.4 3.9 Labor

5.1 5.2 5.4 6.3 Miscellaneous

Since cost is given in multiples of $1000 and production in multiples of 10,000 units, the entries of Care multiples of $10 millions; thusell = 13.2 means $132 million, etc.

Suppose that in a weight-watching program, a person of 185 lb burns 350 cal/hr in walking (3 mph), 500 in bicycling (13 mph), and 950 in jogging (5.5 mph). Bill, weighing 185 lb, plans to exercise according to the matrix shown. Verify the calculations (W=Walking, B= Bicycling, J=Jogging).

W B J

MON 1.0 0 0.5 825 MON

[350]

WED 1.0 1.0 0.5 1325 WED

500 =

FRI 1.5 0 0.5 1000 FRI

950

SAT 2.0 1.5 1.0 2400 SAT

Suppose that the 2004 state of land use in a city of 60 mi²of built-up area is

C: Commercially Used 25% I: Industrially Used 20% R: Residentially Used 55%.

Find the states in 2009, 2014, and 2019, assuming that the transition probabilities for 5-year intervals are given by the matrix A and remain practically the same over the time considered.

FromC From I FromR

[0.7

^0.1

~.2]

^ToC

A= 0.2 0.9 To I

0.1 0 0.8 ToR

CHAP. 7 Linear Determinants. Linear ....,,,q ...,...

A is astochastic matrix,that is, a square matrix with all entries nonnegative and all column sums equal to 1.

Our example concerns aMarkov process.'that is, a process for which the probability of entering a certain state depends only on the last state occupied (and the matrix A), not on any earlier state.

Solution. From the matrix A and the 2004 state we can compute the 2009 state,

c

R

1 ^0.

^{7 .}

²⁵

⁺

^{0.1 .20}

⁺

^{0 .55] lo.}

⁷

l

^{0.1 . 25O.2 . 25++0.9 . 20O· 20++ 0.2 . 550.8 . 55 = 0.20.1}

0.1 0.9

o

o

l r ^25] r ^19.5l

0.2

J

^{l20 = l34.0}

J .

0.8 55 46.5

To explain: The 2009 figure for C equals 25% times the probability 0.7 that C goes into C, plus 20% times the probability 0.1 that I goes into C, plus 55 % times the probability 0 that R goes into C. Together,

25 . 0.7+20 . 0.1+55 . 0 = 19.5 [% J. Also 25 . 0.2+ 20 . 0.9+55 . 0.2= 34 [% ].

Similarly, the new R is 46.5 %. We see that the 2009 state vector is the column vector y= 34.0 46.5JT=Ax =A [25 20 55JT

where the column vector x= [25 20 55JT is the given 2004 state vector. Note that the sum of the entries of yis 100 [%]. Similarly, you may verify that for 2014 and 2019 we get the state vectors

z=Ay= A(Ax) =A^2x= u Az=

43.80 39.15JT 50.660

Answer. In 2009 the commercial area will be 19.5% (11.7 the industrial 34%(2004 and the residential 46.5% (27.9 For 2014 the corresponding are 17.05%,43.80%, and 39.15%. For 2019 they are 16.315%,50.660%, and 33.025%. (In Sec. 8.2 we shall see what happens in the assuming that those probabilities remain the same. In the meantime, can you experiment or guess?)

for 3X 3 Nilpotent matrix, defined

Can you find three 2X 2 .....L..LIIJ"-''''...,...'''

Can you matrices? For m Xnmatrices?

Let

2 ^-1 3 -1 3 0

A= -2 4 -3 0

2 -2 0 0 2

3

-2 2 =[- -2

OJ,

^-1

2 0

of matrices

t1"'1<:1-nnn"lI<:11'"and rouowina are

+

What form does a 3 X 3 matrix have

C"U1mnnpt"r1r>as well assxew-svmmetnc

Can every 3X 3 matrix be two vectors as in 3?

How many different entries

~la,,:w·-~vmrnp,1Inrmatrix have? AnnX n

+ +

~l("f~"Ul.-~'Un1rnp1-rlr>matrix?

rnl,p"t1"'-Y'C' as in Prob. 4 forC"\Trnrnpt1"'"Ir>matrices.

Idemuotent matrix,defined = A. Can you find

four 2 X 2 matrices?

3.

4.

5. Same 6.

lANDREI ANDREJEVITCH MARKOV (1856-1922), Russian mathematician, known for his work in probability theory.

Matrix^{I \ / l l}.11i-1I",11,-~i-lll"'\n

A3.1.

cosine cos -sin

Show that in (a)

appltcanons. We show in this matrices.

the linear

C T

60 120 S

-sine], x=

[Xl],

y = [YI]

cos

e

^{X2 Y2}

An = [cosnO sinnf)

[ ^c~s

^SInaa

[

COS(a + sin(a +

[ COS

e

A=

sin

e

A=

Is this plausib!e (c) Addition

geometry we should have

-sin [cos f3 -sin cos sin f3 cos

- sin(a +

{3)].

cos(a+ f3) Derive from this the addition formulas in

there is no trouble, what is the of

after Three after

CAS Markov Write a program

for a Markov process. Use it to calculate further steps

in 13 of the text. with other

stochastic 3 X3 matrices, also different values.

is a counterclockwise rotation of the CartesianXIX 2-

coordinate system in the about the where

o

^{is the} of rotation.

(b)

28. Concert In a of 100,000

adults, subscribers to a concert series tend to renew their

01l1-'Cr'lA l n t l r - . nwith 90% and persons

not will subscribe for the next season with probatnntv0.2 %. If the present number of subscribers is 1200, can one an increase, decrease, or no

over each of the next three seasons?

outlets and F²in New sell sofas (S), chairs (C), and

tables (T) and

the sales in a certain week be in Prob. 1 columnwise. See

+cC + ... +mM, aA +

(14)

C + ^isC"'{TrnrY"lptrllr'and C - is skew-svmmetnc.

Write C in the form = S+ where S isC"'{TrnrY"lptr-1r'

and and find SandT in terms

of C. in Probs. 11-20 in this form.

A of A, C,"',

of the same size is an of the form

Fr()(lUlCU10ll. In a process, letN mean "no trouble" andT"trouble." Let thei-...."'".. ,.,,,i-,,r.,..,,r\rr-.hl),h111tl~:>C

from one to the next be 0.8 forN -~N,hence 0.2 for N ~ T, and 0.5 for T ~ N, hence 0.5 for T ~ T.

the claims in (11) that akj= ajk for a matrix, and akj = - ajk for a skew-

c ' u r n r Y " l p t r l r 'matrix. Give examples.

Show that for every square matrix C the matrix

where a,"', m are any scalars. Show that if these matrices are square and so is(14);

if are so is(14).

Show that AB with A and if and if A and commute, that is,

(e) Under what condition is the of skew-

C"'{TrnrY"lptrlr'matrices 1;;.!('P"Xl-I;;.VrnmIPtr'lr"!

... rt"""I'" ,,-all intermediate results, calculate the tOllO\VIDJ!

are undefined:

rules. Prove (2) for 2X 2 matrices A = [ajk],

B = [bj k ] ,C = [Cjk],and a scalar.

22. Write AB in Prob. 1 in terms of row and column vectors.

23.

15. Aa,

ABa, ABb, 18. ab, ba, aA,

1.5a+ 3.0b,

Calculate

1l....v-'r,...-Y\r\1C1 1.

24. Find all 2X 2 matrices = [ajk]

that commute with =j + k.

25"TEAM PROJECT" Svmmetrtc and Skew-Svmmetric Matrices. These matrices occur

apnncauons, so it is worthwhile to

研究所考過證明 (86中央環工) (見補充資料)

CHAP. 7 Linear Matrices, Vectors, Determinants. Linear

(d) To visualize a three-

dimensional object with plane faces (e.g., a cube), we may store the position vectors of the vertices with respect to a suitable^{XIX 2X}3-coordinate system (and a list of the connecting edges) and then obtain a two- dimensional image on a video screen by projecting the object onto a coordinate plane, for instance, onto the Xlx2-plane by setting ^X3 == O. To change the appearance of the image, we can a linear transformation on the position vectors stored. Show that a diagonal matrix with main diagonal entries3, 1,~ from an x == [Xj] the new vector y == Dx, where Yl == (stretch in the

by a factor 3),Y2 == ^{X2 Y3} == (con- traction in the What effect would a scalar matrix have?

Rotations space. Explainy ==Ax geometrically whenA is one of the three matrices

[:

0 0

cos () -sin () sin () cos ()

cos<.p 0 -sin^<.p cosljJ -sinljJ 0

0 0 sinljJ cosljJ

o .

sin<.p 0 cos<.p 0 0

What effect would these transformations have in situations such as that described in

We now come to one of the most use of that matrices to

solve of linear We showed in 1 of Sec. how

to the information contained in a of linear a called

the matrix. This matrix will then be used in the linear of

""""-'I •.-...""JL'-J'JUlU.Our to linear is called the Gauss elimination method.

Since this method is so fundamental to linear the student should be alert.

A shorter term for of linear Linear ~VI~rp1TI~

model many and many other areas.

Electrical and markets may serve as

of<:l1"'\1nl1,0<:lt·1n.l'lC

A linear the form

of n ^rmunnwns Xl, ...^,X_n is a set of a.ntll"Jl"Jl ...·..~lI'.,.C' of

is called linear because each variable xj appears in the first power

a.ntlll"Jl"t·l~nof a line. all, ... ,a_{m n}are called the coefficients

of the b_{l , ... ,}b_m on the are also numbers. If all theb_j are zero, then (1) is called a If at least one bj is not zero, then is called a

高斯消去法

擴大矩陣

(1) n個未知數 (2) m個方程式

齊次系統 (bi=0) 非齊次系統

(bi≠0)

係數 必須先按照順序排

好，缺項補為0

x,y,z軸的旋轉

(見前「齊次座標」補充資料)

SEC. 7.3 Linear''If:OT"O''''''''f:O of1-1"'1111 "'I>TI.,...I"\("' Gauss Elimination

A solution of (1) is a set of numbers Xl, ...,Xn that satisfies all the m equations.

A solution vector of (1) is a vector x whose components form a solution of (1). If the system (1) is homogeneous, it always has at least the trivial solutionxI == 0, ...,Xn == 0.

Matrix Form of Linear From the definition of matrix ~1"IIt-",,,,I^..,,,,..,.t-·.~,....

we see that the mt3r1111<:l-t1,n.nC' of (1) may be written as a single vector ....,.... '...IALJL'U'..I...I.

where the coefficient A == [ajkJ is them X n matrix

and and b ==

are column vectors. We assume that the coefficientsajk are not all zero, so that A is

not a zero matrix. Note that has n whereas The

matrix

A==

a_{m n}

is called the The vertical line could be

V..l..L..L.Ll-l-'...\.-D., as we shall do later. It is a reminder that the last column of

A

did not

come from matrix A but came from vector the matrix A.

Note that the

A

^{because it}

contains all the

If^m = n= 2,we have two equations in two unknowns ^{Xl, X2}

If we interpretXl, X2as coordinates in the Xlx2-plane, then each of the two equations represents a straight line, and^{(Xl, X2)}is a solution if and only if the pointP with coordinates^{Xl, x2}lies on both lines. Hence there are three possible cases (see Fig. 158 on next page):

(a) Precisely one solutionifthe lines intersect (b) Infinitely many solutions if the lines coincide (c) No solutionifthe lines are parallel

恆零解→齊次系統獨有

變成AX=C矩陣形式 (一定要會的技巧！)

增廣矩陣 增大矩陣 擴大矩陣

三種解的狀況，及 其對應的幾何情況

Matrices, Vectors, Determinants. Linear " ...

For instance,

Xl+X2=1 2xI -x2=0 Case (a)

Xl+x2=1 2x I+2x2=2

Case (b)

Xl+x₂= 1

xl+x₂=0 Case (c) Unique solution

Infi nitely many solutions

p

the system is homogenous, Case (c) cannot happen, because then those two straight lines pass through the origin, whose coordinates (0, 0) constitute the trivial solution. Sirnilarly, our present discussion can be extended from two equations in two unknowns to three equations in three unknowns. We give the geometric interpretation of three possible cases concerning solutions in Fig. 158. Instead of straight lines we have planes and the solution depends on the positioning of these planes in space relative to each other. The student may wish to come with some specific examples.

5 3

Its -:lH4:TtTIpntpr! matrix is

2 -30.

+ +

2 -26

may have solution. This leads to such have solution? Under what conditions does it have

nr~:l>01lCP."uone solution? it has more than one how can we characterize the set

of all solutions ? We shall consider such in Sec. 7.5.

hr"""'1a"1Cl>1I'" let us discuss method for linear~V;"~I.LliI~.

The Gauss elimination method can be motivated as follows. Consider a linear that

is upper such as

0A-r-rpc~n{"\nti1In^0- coefficient matrix lie

"' ... ""' ...~... ! Then we can solve the

J.ULlIl-\./\.-IU(..IlU.'-J'J.J.for the x2 ==

and then work uU"",,,JL'1l.. V'V UJL'u.c substitutmzx2 == -2 into the first and

Aht'-::nnlno-Xl ==

!

^{(2 - ==}

!

^{(2 - 5 . == 6. This} us the idea of first-rarilllllf~lI1Y".n-

to form. For instance,let the be

No solution

We leave the first as is. We eliminate x1 from the second to For this we add twice the first and we do the same

恰有一解 (交於一點)

無解 (平行線) 無限多解

(重合線) 三個空間中的平面，

可能的交會狀況

反代換

高斯消去法的目的：化簡成三角矩陣形式

反代換求解

注意步驟！

(1)寫出增廣矩陣

2Row1+Row2

SEC. 7.3 Linear .... ',.-_... .-... ....'-! ....\A''-,_, ,....Gauss Elimination

operation on therowsof the augmented matrix. This -30

+

2 . 2, that is,

+

+

2

13x2== -26 Row

+ [~

¹³⁵

where Row

is theGauss 2Orn1<:11"11"'1""0

which back substitution now Since a linear

eumtnauon can be done

We do this in the next exampte,0rlt1I1t"\h<:101'7'11"'!ICTthe matrices

the OrIl111<:1 ....1I"'....\1:'behind in order not to track.

Solve the linear system

80.

(KVL). In any closed loop, the sum of all voltage drops equals the impressed This is the system for the unknown currents

Xl = is-X2 = i2 , X3 =i3in the electrical network in Fig. 159. To obtain it, we label the currents as shown, choosing directions arbitrarily; if a current will come out negative, this will simply mean that the current flows against the direction of our arrow. The current entering each battery will be the same as the current leaving it.

The equations for the currents result from Kirchhoff's laws:

Kirchhoff's Current Law (KCL). At any point of a circuit, the sum of the inflowing currents equals the sum of the outflowing currents.

Kirchhoff's Voltage electromotive force.

Node Pgives the first equation, node Q the second, the right loop the third, and the left loop the fourth, as indicated in the figure.

NodeP: -,- i₂+ i₃= 0

~

^{90 V NodeQ: -i1+ i2- i3= 0}

80

Right loop: + =90

Left loop: 20i₁+10i₂ =80

Network in 2and the

Solution Gauss This system could be solved rather quickly by noticing its particular form. But this is not the point. The point is that the Gauss elimination is systematic and will work in general,

(2)進行高斯消去法，

變成三角矩陣形式 (進行Row運算)

(4)反代換求解 (3)寫出化簡後 之方程式

(1)注意對齊性 (2)缺項補0

利用高斯消去法求解！