Mathematical Methods for Physicists: A concise introduction - Site Map

FOUR BASIC ALGEBRA OPERATIONS FOR MATRICES
When you ®rst run into Eq. (3.7), the rule for matrix multiplication, you might
ask how anyone would arrive at it. It is suggested by the use of matrices in
connection with linear transformations. For simplicity, we consider a very simple
case: three coordinates systems in the plane denoted by the x 1 x 2 -system, the y 1 y 2 -
system, and the z 1 z 2 -system. We assume that these systems are related by the
following linear transformations
x 1 ˆ a 11 y 1 ‡ a 12 y 2 ; x 2 ˆ a 21 y 1 ‡ a 22 y 2 ; …3:8†
y 1 ˆ b 11 z 1 ‡ b 12 z 2 ; y 2 ˆ b 21 z 1 ‡ b 22 z 2 : …3:9†
Clearly, the x 1 x 2 -coordinates can be obtained directly from the z 1 z 2 -coordinates
by a single linear transformation
x 1 ˆ c 11 z 1 ‡ c 12 z 2 ; x 2 ˆ c 21 z 1 ‡ c 22 z 2 ; …3:10†
whose coecients can be found by inserting (3.9) into (3.8),
Comparing this with (3.10), we ®nd
or brie¯y
x 1 ˆ a 11 …b 11 z 1 ‡ b 12 z 2 †‡a 12 …b 21 z 1 ‡ b 22 z 2 †;
x 2 ˆ a 21 …b 11 z 1 ‡ b 12 z 2 †‡a 22 …b 21 z 1 ‡ b 22 z 2 †:
c 11 ˆ a 11 b 11 ‡ a 12 b 21 ; c 12 ˆ a 11 b 12 ‡ a 12 b 22 ;
c 21 ˆ a 21 b 11 ‡ a 22 b 21 ; c 22 ˆ a 21 b 12 ‡ a 22 b 22 ;
c jk ˆ X2
iˆ1
a ji b ik ; j; k ˆ 1; 2; …3:11†
which is in the form of (3.7).
Now we rewrite the transformations (3.8), (3.9) and (3.10) in matrix form:
~X ˆ ~A ~Y; ~Y ˆ ~B ~Z; and ~X ˆ ~C ~Z;
where
~X ˆ x1
x 2
!
; ~Y ˆ y1
y 2
!
; ~Z ˆ z1
z 2
!
;
!
!
~A ˆ a11 a 12
; ~B ˆ b11 b 12
; C ::: !
ˆ c11 c 12
:
a 21 a 22 b 21 b 22 c 21 c 22
We then see that ~C ˆ ~A ~B, and the elements of ~C are given by (3.11).
Example 3.5
Rotations in three-dimensional space: An example of the use of matrix multiplication
is provided by the representation of rotations in three-dimensional
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