INTRODUCTION TO STATISTICS AND ECONOMETRICS
MATRIX OPERATIONS
Equality. If A and В are matrices of the same size and A = {аф and В = {Ьф, then we write Ip = В if and only if at] = by for every і and j.
Addition or subtraction. If A and В are matrices of the same size and A = {аф and В = {Ьф, then A ± В is a matrix of the same size as A and В whose i, jth element is equal to al} ± by. For example, we have
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«11 |
«12 |
Ч- |
'*11 |
*12 |
«11 ±*H |
dl2±bi2 |
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«21 |
«22 |
*21 |
*22 |
«21 ± *21 |
0>22 — ^22 |
Scalar multiplication. Let A be as in (11.1.1) and let c be a scalar (that is, a real number). Then, we define cA or Ac, the product of a scalar and a matrix, to be an n X m matrix whose i, jth element is сац. In other words, every element of A is multiplied by c.
Matrix multiplication. Let A be an n X m matrix {аф as in (11.1.1) and let В be an m X r matrix {Ьф. Then, C = AB is an n X r matrix whose i, j'th element Сц is equal to 1ф=їаікЬк]. From the definition it is clear that matrix multiplication is defined only when the number of columns of the first matrix is equal to the number of rows of the second matrix. The exception is when one of the matrices is a scalar—the case for which multiplication was previously defined. The following example illustrates the definition of matrix multiplication:
«11^11 + «12*21 + «13*31 «11*12 + «12 *22 + «13^32
«21 *11 + «22 *21 + «23*31 «21 *12 + «22 *22 + «23 *32
If A and В are square matrices of the same size, both AB and BA are defined and are square matrices of the same size as A and B. However, AB and BA are not in general equal. For example,
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'і і |
1 rH GO 1_ |
5 5 |
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|
0 1 |
1 2_ |
1 2 |
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|
’з l‘ |
"l 2 |
’з І |
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|
----- 1 CM 1—1 __ 1 |
1---- О )—‘ 1____ |
1 4 |
In describing AB, we may say either that В is premultiplied by A, or that A is postmultiplied by B.
Let A be an n X m matrix and let I„ and Im be the identity matrices of size n and m, respectively. Then it is easy to show that InA = A and AIm = A.
Let a' be a row vector (<q, a%, . . . , an) and let b be a column vector such that its transpose b' = (b, b%,. . . , b„). Then, by the above rule of matrix multiplication, we have a'b = Е"=1аД, which is called the vector product of a and b. Clearly, a'b = b'a. Vectors a and b are said to be orthogonal if a'b = 0. The vector product of a and itself, namely a'a, is called the inner product of a.
The proof of the following useful theorem is simple and is left as an exercise.
THEOREM 11.2.1 If AB is defined, (AB)' = B'A'.
