Mathematical Tools for Physics

7—Operators and Matrices 182

7.4 Matrix Multiplication
How do you multiply two matrices? There’s a rule for doing it, but where does it come from?
The composition of two functions means you first apply one function then the other, so

h=f◦g means h(~v) =f

(

g(~v)

)

(20)

I’m assuming that these are vector-valued functions of a vector variable, but this is the general definition of
composition anyway. Iffandgare linear, does it follow thehis? Yes, just check:

h(c~v) =f

(

g(c~v)

)

=f

(

cg(~v)

)

=cf

(

g(~v)

)

, and

h(~v 1 +~v 2 ) =f

(

g(~v 1 +~v 2 )

)

=f

(

g(~v 1 ) +g(~v 2 )

)

=f

(

g(~v 1 )

)

+f

(

g(~v 2 )

)

What are the components ofh? Again, use the definition and plug in.

h(~ei) =

∑

k

hki~ek=f

(

g(~ei)

)

=f

(∑

j

gji~ej

)

=

∑

j

gjif

(

~ej

)

=

∑

j

gji

∑

k

fkj~ek

and now all I have to do is equate the corresponding coefficients of~ek.

hki=

∑

j

gjifkj or more conventionally hki=

∑

j

fkjgji (21)

This is in the standard form for matrix multiplication, recalling the subscripts are ordered asfrcfor row-column.




h 11 h 12 h 13

h 21 h 32 h 23

h 31 h 32 h 33



=





f 11 f 12 f 13

f 21 f 32 f 23

f 31 f 32 f 33









g 11 g 12 g 13

g 21 g 32 g 23

g 31 g 32 g 33



 (22)

The computation ofh 12 from Eq. ( 21 ) is




h 11 h 12 h 13

h 21 h 22 h 23

h 31 h 32 h 33



=





f 11 f 12 f 13

f 21 f 22 f 23

f 31 f 32 f 33









g 11 g 12 g 13

g 21 g 22 g 23

g 31 g 32 g 33





−→ h 12 =f 11 g 12 +f 12 g 22 +f 13 g 32