7. The Elements of Vector Spaces
10.3 Matrices and Their Algebra 136
In Section 10.1 we saw that a linear transformation T:m,3---+ m.3can be represented (that is, is completely determined) by 9 numbers arranged in a 3x3 array. In this section we will study such arrays, which are called matrices. We will return to the connection between matrices and linear transformations in the next chapter.
DEFINITION: A rectangulararrayofnumberscomposed ofm rows and n columns
[ :~~ :~:
A=
am,: ~m,2
is calledanmxn matrix(read "m byn matrix").1
The elementsai,1> ai,2, ... ,ai,nform the i-th row ofA, and the elements
am,)
form the j-th column of A. We will often write
for A, or simply
A=(ai,)
when m and n are understood from context. Note that the order of the subscripts is important; the first subscript denotes the row, and the second subscript the column,towhich an entry belongs.
Just as with vectors inm,n, two matrices are equal if and only if they have the same entries. That is:
1We also say that the matrix A is of, or has, sizemxn.
E(3, 2)=
10.3Matrices and Their Algebra 137 DEFINITION: IfA=(ai,j) and B=(bi,j) are mx n matrices, then A=Bifandonly ifai,j=bi,j fori =1, 2, ... , m andj =1, ... , n.
Our study of linear transfonnations suggests the following definitions.
DEFINITION: IfA=(aU) and B=(bi,j) arem xn matrices, their sum, A+B,is thematrixC=(ci,), whereci,j=ai,j+bi,j,i =1,2, ... , m andj =1, 2, ... , n.
DEFINITION: IfA=(ai,) is an m xn matrix, and r is a number then rA, the scalar multiple of Abyr is the matrixC=(Ci,), where Ci,j=rai,j, andi =1, ... , m andj=1, ... , n.
The following result is a routine verification of definitions:
PROPOSITION 10.3.1: Thematricesofsizemxnfonnavector space under the operations ofmatrix additionandscalar multiplication. We denote this vector spacebyMatm,n. D
The dimension ofthe vector space Matm,n is not hard to compute. We take our lead from the method we used to show that dim(IRn) = n.
Introduce the mxnmatrix E(r,s) =(ei,j)by the requirement { I i =r,j=s,
ei,j= 0 otherwise.
For example, the 6 x 4 matrix E(3, 2) is
o 000
000 0
o 1 0 0
000 0
o 0 0 0 o 000
The matrices E(r,s) are called elementary matrices. Itis then a routine verification to prove:
PROPOSITION 10.3.2: The vectors
{E(r,s)Ir =1,2, ... , m; s =1,2, ... , n}
form abasisforMatm,n. Therefore, dim(Matm,n)=mn. D
Recall that for vector spaces 'l! and 'W we introduced (see Theorem 8.5.6) the vector spaceL('l!, 'W)oflinear transfonnations from'l!to'W.
If'l!and 'Ware finite-dimensional, then we will see that the innocent looking Proposition 10.3.2implies the useful fact that dim(L('l!,'W))=
dim('l!) . dim('W). We record this, since the symmetry it implies is a priori not at all clear.
COROLLARY 10.3.3: If'll and W are finite-dimensional vector spaces then dim(L('ll, Wằ =dim('ll)ãdim(W) =dim(L(W,'llằ. 0
EXAMPLE 1: Here are some examples ofmatrix addition and scalar multiplication
[
3 1 2 0 -2 -1
1-1 4+3]
0-9 -1+4 -1+6 0+1
[3 1] [12 4]
4 7 4 = 28 16.
6 -4 24 -16
The discussion of the matrix of the composition of two linear transfor- mations in IR3from Section 10.1 suggests that the following might be a useful definition.
DEFINITION:IfA=(ai,j)isanmxn matrix andB=(bi,j)isannxp matrix theirmatrix product2 Aã Bis the m xp matrixA . B=(Ci,)
where
n
cã .1,) ='""~aã1,kbk .,)
k=I
fori =1, ... , m;j =1, ... , p.
Thus the entry of the i-th row and j-th column of the product A . B is obtained by taking the i-th row
of the matrix A and the j-th column of the matrix B
[ bI,j]
b2,j
. ,
bn,j
2The matrix product is also denoted by AB, i.e., without theã if no confusion can arise.
10.3 Matrices and Their Algebra 139 multiplying the corresponding entries together, and adding the result- ing products, Le.,
ai,lb1,j+ai,2 b 2,j+ ... +ai,kbk,j+ ... +ai,nbn,j,
i =1,2, ... , m;j=1,2, ... , p.
Note that for the product ofAand Bto be defined, the number of columns ofAmust be equal to the number of rows ofB. Thus theorderin which the product of A and B is taken is very important, for A . B can be defined without the product B . A being defined at all, much less being equal to Aã B.
EXAMPLE 2: Compute the matrix product
[123].[:].
SOLUTION: Note the answer must be a 1 x 1 matrix.
[123]ã[i]=[1.4+2.5+3.6 1 =[32].
The product in the reverse order,
[4]5 . [1 2 3] = [4 85 10 15 ,12]
6 6 12 18
does not even have the same size: It is 3 x 3 and not 1 x 1. So the products in different orders are certainly not equal.
EXAMPLE 3: Compute the matrix product
[0 0 0 0 0 0~ ~ ~]. [~ ~ ~].
SOLUTION:
EXAMPLE 4: Let
A=[~ ~:] and B=U ~ i:]
Calculate the product Aã B.
SOLUTION:We have
[~ : :]. [~ ~ ~ :]
[1+2+3 2+4+6
= 4 + 5 + 6 8 + 10 + 12 [ 6 12 18 24]
= 15 30 45 60 .
3+6+9 4+8+12]
12 + 15 + 18 16 + 20 + 24
Note that the product B . A isnotdefined.
DEFINITION: Amatrix Ais saidto be squareof size n ifithasn rowsandn columns (that is, thenumber ofrows equals thenumber of columns equals n).
If A and B are square matrices of sizen,then the productsABandBA are both defined and are square matrices of sizen. However, they may not beeq~al.
EXAMPLE 5: Let
A= [~ ~] and B= [~ ~].
Compute the matrix productsAB and BA.
SOLUTION: We have
and we see thatAB :j:. BA.
As the preceding example shows, even ifAB and BA are defined, we should not expect thatAB=BA.
NOTE: If A is a square matrix, thenAAis defined and is denoted by A2• Similarly,
A .. ãA
-n--+
is defined and denotedAn.
EXAMPLE 6: Let
CalculateA2.