Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
Four basic algebra operations for matrices
Equality of matrices
Two matrices ~A ˆ…a jk † and ~B ˆ…b jk † are equal if and only if ~A and ~B have the
same order (equal numbers of rows and columns) and corresponding elements are
equal, that is
a jk ˆ b jk for all j and k:
Then we write
~A ˆ ~B:
Addition of matrices
Addition of matrices is de®ned only for matrices of the same order. If ~A ˆ…a jk †
and ~B ˆ…b jk † have the same order, the sum of ~A and ~B is a matrix of the same
order
~C ˆ ~A ‡ ~B
with elements
c jk ˆ a jk ‡ b jk :
We see that ~C is obtained by adding corresponding elements of ~A and ~B.
…3:2†
Example 3.1
If
~A ˆ 2 1 4
; ~B ˆ 3 5 1
3 0 2
2 1 3
hen
~C ˆ ~A ‡ ~B ˆ 2 1 4
!
‡ 3 5 1
!
!
2 ‡ 3 1‡ 5 4‡ 1
ˆ
3 0 2 2 1 3 3 ‡ 2 0‡ 1 2 3
ˆ 5 6 5
!
:
5 1 1
From the de®nitions we see that matrix addition obeys the commutative and
associative laws, that is, for any matrices ~A, ~B, ~C of the same order
~A ‡ ~B ˆ ~B ‡ ~A; ~A ‡…~B ‡ ~C† ˆ…~A ‡ ~B†‡ ~C: …3:3†
Similarly, if A ~ ˆ…a jk † and ~B ˆ…b jk ) have the same order, we de®ne the di€erence
of ~A and ~B as
~D ˆ ~A ~B
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