Mathematical Methods for Physicists: A concise introduction - Site Map

FOUR BASIC ALGEBRA OPERATIONS FOR MATRICES
with elements
d jk ˆ a jk b jk :
…3:4†
Multiplication of a matrix by a number
If ~A ˆ…a jk † and c is a number (or scalar), then we de®ne the product of ~A and c as
c ~ A ˆ ~Ac ˆ…ca jk †;
…3:5†
we see that c ~ A is the matrix obtained by multiplying each element of ~ A by c.
We see from the de®nition that for any matrices and any numbers,
c… ~ A ‡ ~B† ˆc ~ A ‡ c ~B; …c ‡ k† ~ A ˆ c ~ A ‡ k ~ A; c…k ~ A†ˆck ~ A: …3:6†
Example 3.2
7 a b c
ˆ
d e f
7a 7b 7c
7d 7e 7f
Formulas (3.3) and (3.6) express the properties which are characteristic for a
vector space. This gives vector spaces of matrices. We will discuss this further in
Chapter 5.
:
Matrix multiplication
The matrix product ~A ~B of the matrices ~A and ~B is de®ned if and only if the
number of columns in A ~ is equal to the number of rows in ~B. Such matrices
are sometimes called `conformable'. If ~A ˆ…a jk † is an n s matrix and
~B ˆ…b jk † is an s m matrix, then ~A and ~B are conformable and their matrix
product, written ~C ˆ ~A ~B, isann m matrix formed according to the rule
c ik ˆ Xs
jˆ1
a ij b jk ; i ˆ 1; 2; ...; n k ˆ 1; 2; ...; m: …3:7†
Consequently, to determine the ijth element of matrix ~ C, the corresponding terms
of the ith row of ~A and jth column of ~B are multiplied and the resulting products
added to form c ij .
Example 3.3
Let
0 1
3 5
2 1 4 B C
~A ˆ
; ~B ˆ @ 2 1 A
3 0 2
4 2
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