Mathematical Tools for Physics

7—Operators and Matrices 175

For two vectors to be equal, the corresponding coefficients of~e 1 ,~e 2 ,etc.must match; their respective components
must be equal.

uk=

∑

i

vifki, usually written uk=

∑

i

fkivi (8)

so that in the latter form it starts to resemble what you may think of as matrix manipulation. frow,columnis the
conventional way to write the indices, and multiplication is defined so that the following productmeansEq. ( 8 ).




u 1

u 2

u 3



=





f 11 f 12 f 13

f 21 f 22 f 23

f 31 f 32 f 33









v 1

v 2

v 3



 (9)





u 1

u 2

u 3



=





f 11 f 12 f 13

f 21 f 22 f 23

f 31 f 32 f 33









v 1

v 2

v 3



 is u 1 =f 11 v 1 +f 12 v 2 +f 13 v 3 etc.

And this is the reason behind the definition of how to multiply a matrix and a column matrix. The order in which
the indices appear is the conventional one, and the indices appear in the matrix as they do because I chose the
order of the indices the way that I did in Eq. ( 6 ).

Components of Rotations
Apply this to the first example, rotate all vectors in the plane through the angleα. I don’t want to keep using the
same symbolfforevery function that I deal with, so I’ll call this functionRinstead, or better yetRα. Rα(~v)
is the rotated vector. Pick two perpendicular unit vectors for a basis. You may call themxˆandyˆ, but again I’ll
call them~e 1 and~e 2. Use the definition of components to get

Rα(~e 2 )

~e 2

cosα

α

Rα(~e 1 )

sinα

~e 1

Rα(~e 1 ) =

∑

k

Rk 1 ~ek

Rα(~e 2 ) =

∑

k

Rk 2 ~ek

(10)