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,
|
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