product

mathematical description

in finite dimensions

physical description

M × M M

the composition of 2 multiplication operators is a multiplication operator

the product of 2 diagonal matrices is a diagonal matrix

stacking 2 transmitting filters effectively creates a new filter

L × L* R

evaluate a functional on a vector to get a scalar – the response

a row vector × a column vector is a scalar – the response

light hits a detector and generates a response

L × M L

a multiplication operator acts on a vector to create a vector

a row vector × a diagonal matrix is a row vector

light passes through a filter and emerges with a different spectrum

M × L* L*

an operator on L, followed by a functional on L,
is a functional on L

a diagonal matrix × a column vector is a column vector

putting a transmitting filter in front of a light responder, effectively creates a new light responder

M × M* R

evaluate a functional on a vector to get a scalar – the response

the product of 2 diagonal matrices is a diagonal matrix; the response is then the trace of that matrix

a scanner (M*) (with both a light source and a light responder) responds to a material (M) placed in the scanner

Table C.2 The 5 natural products