经济数学讲义(研究生版)

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1. Vector
A vector is an ordered sequence of numbers:
æ ö ÷ ç ç ÷ 1÷ ç ÷ ç ÷ ç ÷ ÷ ç ÷ ç ÷ ç ÷ ç ÷ ç ÷ ÷ ç ÷ ç ÷ ç ÷ ç è n÷ ø
x
x = ( xi )n = Î n , x
rank ( A) = the maximum number of independent column vectors = the size of the largest invertible square submatrix of A. æ2 -1 3ö ÷ ç ÷ ç ÷ ÷ 4 2 5 . We have rank ( A) = 2. Example 1.5. Consider matrix A = ç ç ÷ ç ÷ ç ÷ ç è2 -1 4÷ ø
For A Î n´n , denote the trace of A as
tr ( A) º a11 + a22 + + ann .
Theorem 1.7. • • • • •
tr ( cA) = c ⋅ tr ( A). tr ( A¢) = tr ( A). tr ( A + B ) = tr ( A) + tr ( B ). tr ( AB ) = tr ( BA). tr (T -1 AT ) = tr ( A).
(1.1)
Given a vector in (1.1), we define its transpose and denote it as x ¢ or x T . There is also a zero vector 0 Î n . Given two vectors a , b Î n , we can define their summation a + b, subtraction a - b, and multiplication a ¢b, a , b , a ⋅ b. These operations can be intuitively shown in the Euclidean space n . Define the length of a vector as x and call it the norm. We can define the distance of any two points in n by the norm. Proposition 1.1. For vectors a, b, c Î n , we have (a) Associative law of summation: ( a + b) + c = a + (b + c ). (b) Commutative law of summation: a + b = b + a. (c) Commutative law of multiplication: a ⋅ b = b ⋅ a. (d) Distributive law: a ⋅ (b + c ) = a ⋅ b + a ⋅ c.
(1.2)
We often denote the matrix in (1.2) as A = ( aij ) m´n and it is said to be of dimension m ´ n. Example 1.2. A linear equation system can be written as Ax = d . A special matrix is the zero matrix: 0 Î m´n . When m = n, we have a square matrix. A special square matrix is the identity matrix:
A-1 =
ห้องสมุดไป่ตู้
1 * A. | A|

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Example 1.4. We have
æa b ö ö d -b÷ 1 æ ÷ ç ç = . ÷ ÷ ç ç ÷ ÷ çc d ÷ ad - bc ç è ø è-c a ÷ ø
Theorem 1.5. If A, B Î n´n are invertible, then • ( A-1 )-1 = A, • ( AB )-1 = B-1 A-1. Given a matrix A Î m´n , denote its transpose as A¢ or AT .
Math in Economics
Chapter 1. Linear Algebra
We focus on matrix only. Good references: Sydsæter (2005, Chapter 1), Chiang (1984, Chapters 4 and 5), and Greene (1993, Chapter 2).
.
Theorem 1.3. For any square matrix A = ( aij ) Î n´n and any i and j, we have
(1) (2)
åa
k =1 n k =1
n
ik
Cik = å akjCkj = A , for any i, j;
k =1 n
n
åa
ik
C jk = å akiCkj = 0, for any i, j, i ¹ j.
For a matrix A Î m´n , if the maximum number of linearly independent row vectors is r, then A is of rank r, denoted as rank ( A) = r. Theorem 1.8. For any A Î m´n ,
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We can also multiply a vector by a number: la.
b Î n is a linear combination of a1, …, am Î n , if there exist l1, …, lm Î s.t.
b = l1a1 + + lmam .
k =1
For A Î n´n , if $ B Î n´n s.t.
AB = BA = I ,
then A is invertible or nonsingular, and denote B as A-1. The inverse is unique and we need only either AB = I or BA = I to hold. Theorem 1.4. A Î n´n is invertible | A |¹ 0. In this case,
Am´n + Bm´n , Am´n - Bm´n , Am´n Bn´k .
We can also multiply a matrix by a number: l A. Vectors are treated as special matrices. Theorem 1.1. When the operations are feasible, we have (a) Associative law of summation: ( A + B ) + C = A + ( B + C ). (b) Associative law of multiplication: A( BC ) = ( AB )C.
I n Î n´n .
Given two matrices A and B, we may define their summation A + B, subtraction A - B, and multiplication AB, but the following rules are imposed:
and the cofactor of aij is
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Cij º (-1)i + j M ij .
The adjoint of A :
æ ç ç ç ç ç ç ç ç ç ç ç ç ç ç è ö C11 C1n ÷ ÷ ÷ ÷ T
A* º

Cn 1 C
÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ ÷ nn ø
l1a1 + l2a2 + + lmam = 0.
Vectors which are not linearly dependent are linearly independent. Vectors are linearly dependent iff one of them is a linear combination of the rest. Example 1.1. Any two linearly independent vectors in 2 can span the whole space 2 . This can be shown graphically.
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Theorem 1.6. For the transpose operation, we have • • • • •
( A¢)¢ = A;
( A + B )¢ = A¢ + B¢;
( AB )¢ = B ¢A¢; ( cA)¢ = cA¢; ( A¢)-1 = ( A-1 )¢.
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(c) Commutative law of summation: A + B = B + A. (d) Distributive law: A( B + C ) = AB + AC, ( B + C ) A = BA + CA. The commutative law of multiplication fails for matrices: AB ¹ BA. A vector x Î n can be treated as matrix x Î n´1. If so, all the rules and properties for vectors can be derived from the rules and properties for matrices. That is, vectors are special matrices. Example 1.3. Given two vectors a Î m and b Î n , derive ab¢. Let A be the determinant of a square matrix A. Theorem 1.2. For any A, B Î n´n , we have | AB |=| A || B |. Given A in (1.2) with m = n, the minor of aij is
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2. Matrix
A matrix is an ordered sequence of column vectors or row vectors:
æ ç ç 11 ç ç ç ç ç ç ç ç ç ç ç ç è m1
a
A = (aij )m´n = a
÷ a1n ö ÷ ÷ ÷ ÷ ÷ m´n ÷ ÷Î ÷ . ÷ ÷ ÷ ÷ ÷ ÷ amn ÷ ÷ ø
Define the span of vectors a1, …, am Î n as
span(a1, , am ) = {all linear combinations of vectors a1, , am } .
Vectors a1, a2 , , am Î n are linearly dependent if there exist l1, l2 , …, lm Î , not all 0, s.t.
a11 M ij º

a1, j-1
a1, j +1

a1n ,
ai-1,1 ai-1, j-1 a n1 an, j-1
ai-1, j +1 ai-1, n an, j +1 ann
ai +1,1 ai +1, j-1 ai +1, j +1 ai +1, n