伍德里奇《计量经济学导论--现代观点》2

΄ ΅ A ϭ
2 Ϫ4
Ϫ1 5
7 0
(D.1)
where a13 ϭ 7. The shorthand A ϭ [aij] is often used to define matrix operations.
DEFINITION D.2 (Square Matrix) A square matrix has the same number of rows and columns. The dimension of a square matrix is its number of rows and columns.
Given any real number ␥ (often called a scalar), scalar multiplication is defined as ␥A ϵ [␥aij], or
΄ ␥a11 ␥a21 ␥A ϭ ...
␥a12 ␥a22
... ...
΅␥a1n
␥a2n .
␥am1 ␥am2 . . . ␥amn
For example, if ␥ ϭ 2 and A is the matrix in equation (D.1), then
΄ ΅ ␥A ϭ
4 Ϫ8
Ϫ2 10
14 0.
Matrix Multiplication
To multiply matrix A by matrix B to form the product AB, the column dimension of A must equal the row dimension of B. Therefore, let A be an m ϫ n matrix and let B be an n ϫ p matrix. Then matrix multiplication is defined as
΄a11 0 A ϭ ...
0 a22
0 0
... ...
΅0
0 .
0 0 0 . . . ann
DEFINITION D.5 (Identity and Zero Matrices) (i) The n ϫ n identity matrix, denoted I, or sometimes In to emphasize its dimension, is the diagonal matrix with unity (one) in each diagonal position, and zero elsewhere:
΄͚ ΅ n
AB ϭ
aikbkj .
kϭ1
In other words, the (i, j)th element of the new matrix AB is obtained by multiplying each element in the ith row of A by the corresponding element in the jth column of B and adding these n products together. A schematic may help make this process more transparent:
where aij represents the element in the ith row and the jth column. For example, a25 stands for the number in the second row and the fifth column of A. A specific example of a 2 ϫ 3 matrix is
Summary of Matrix Algebra
where, by the definition of the summation operator in Appendix A,
n
͚ aikbkj ϭ ai1b1j ϩ ai2b2 j ϩ … ϩ ainbnj.
kϭ1
For example,
΄ ΅ ΄ ΅ ΄ ΅ 2 Ϫ1 0 Ϫ4 1 0
PROPERTIES OF MATRIX MULTIPLICATION: (1) (␣ ϩ ␤)A ϭ ␣A ϩ ␤A; (2) ␣(A ϩ B) ϭ ␣A ϩ ␣B; (3) (␣␤)A ϭ ␣(␤A); (4) ␣(AB) ϭ (␣A)B; (5) A ϩ B ϭ B ϩ A; (6) (A ϩ B) ϩ C ϭ A ϩ (B ϩ C); (7) (AB)C ϭ A(BC); (8) A(B ϩ C) ϭ AB ϩ AC; (9) (A ϩ B)C ϭ AC ϩ BC; (10) IA ϭ AI ϭ A; (11) A ϩ 0 ϭ 0 ϩ A ϭ A; (12) A Ϫ A ϭ 0; (13) A0 ϭ 0A ϭ 0; (14) AB BA, even when both products are defined.
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Appendix D
Summary of Matrix Algebra
This appendix summarizes the matrix algebra concepts, including the algebra of probability, needed for the study of multiple linear regression models using matrices in Appendix E. None of this material is used in the main text.
D.1 BASIC DEFINITIONS
DEFINITION D.1 (Matrix) A matrix is a rectangular array of numbers. More precisely, an m ϫ n matrix has m rows and n columns. The positive integer m is called the row dimension, and n is called the column dimension.
We use uppercase boldface letters to denote matrices. We can write an m ϫ n matrix generically as
΄a11 a21 A ϭ [aij] ϭ ...
a12 a22
a13 a23
... ...
΅a1n
a2n
am1 am2 am3 . . . amn
΄A ϩ B ϭ
a11 ϩ b11 a21 ϩ b21
...
a12 ϩ b12 a22 ϩ b22
... ...
am1 ϩ bm1 am2 ϩ bm2 . . .
΅a1n ϩ b1n
a2n ϩ b2n .
amn ϩ bmn
745
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Appendix D
1 0 0 ... 0
΄0
I ϵ In ϵ ...
1
΅ 0 . . . 0 ... .
0 0 0 ... 1
(ii) The m ϫ n zero matrix, denoted 0, is the m ϫ n matrix with zero for all entries. This need not be a square matrix.
Matrix addition, scalar multiplication, and matrix multiplication can be combined in various ways, and these operations satisfy several rules that are familiar from basic operations on numbers. In the following list of properties, A, B, and C are matrices with appropriate dimensions for applying each operation, and ␣ and ␤ are real numbers. Most of these properties are easy to illustrate from the definitions.
D.2 MATRIX OPERATIONS
Matrix Addition
Two matrices A and B, each having dimension m ϫ n, can be added element by element: A ϩ B ϭ [aij ϩ bij]. More precisely,
The last property deserves further comment. If A is n ϫ m and B is m ϫ p, then AB is defined, but BA is defined only if n ϭ p (the row dimension of A equals the column dimension of B). If A is m ϫ n and B is n ϫ m, then AB and BA are both defined, but they are not usually the same; in fact, they have different dimensions, unless A and B are both square matrices. Even when A and B are both square, AB BA, except under special circumstances.