How can an engineer make use of commutative diagram?

Question

I am in engineering and I have being awed by commutative diagram my entire life. But I do not see the purpose in knowing commutative diagram as arises in linear algebra?

Is there something in engineering that can be describe by commutative diagram? For example, operators such as laplace transform and fourier transform to map stuff to different spaces? Or expectation operator...

Thanks

L.Z. Wong · Accepted Answer · 2015-03-05T06:55:43.117

The reason why we can use the Fourier transform to solve differential equations is because the following diagram commutes (from the same Wikipedia article):

enter image description here

The commutative diagram is simply a reasoning device that helps clarify the relationship between certain processes/functions. Often, the fact that a diagram commutes means we can express the same result in two (or more) ways.

Each of these ways might have its own advantages or disadvantages, but knowing that they give the same result (i.e. the diagram commutes) gives us the freedom to choose the method that best suits our purposes.

In simple cases (like the Fourier transform), a commutative diagram probably just tells us something we already know. However, in more complex systems/situations, drawing a commutative diagram might help you see/make use of/make sense of relationships that you might have otherwise missed.

Mariano Suárez-Álvarez · Answer 2 · 2015-02-28T10:05:51.960

4

A commutative diagram is simply a graphic device to represent the fact that two or more compositions of functions are equal.

There is no reason why an engineer will not have two compositions of functions and want to express that they are equal.

There is absolutely nothing mysterious in commutative diagrams. Sometimes they are a bit complicated, like

enter image description here

or even much more, but that is just because this diagram represents the fact that quite a few compositions are equal: each arrow represents a function, each path in the diagram represents the composition of the functions corresponding to the arrows it involves, and the commutativity of the diagram means simply that every time you pick two paths from one of the letters to another, the corresponding compositions are equal.

edited Feb 28 '15 at 10:05

answered Feb 28 '15 at 09:59

Mariano Suárez-Álvarez

135,076

5

In short, how can an engineer make use of commutative diagram? Exactly in the same way everybody else can. – Mariano Suárez-Álvarez Feb 28 '15 at 10:08
Can you please show me what are the As and Bs and the lower case a,b on the line? – Olórin Feb 28 '15 at 17:47
The As and the Bs are sets, the arrows denote functions between them, the $a$s and $b$s that appear here are names for some of those functions (whooever drew this diagram did not need to name all arrows for his purposes) – Mariano Suárez-Álvarez Feb 28 '15 at 17:59

score 2 · Answer 3 · answered Mar 09 '15 at 02:06

Here is a simple example from image processing (software engineering).
From an 8-bit grayscale image, one can compute the mean graylevel - a real number between 0 and 255.
One can also compute the histogram - an integer vector of length 256.
Note that it is also possible to compute the mean graylevel from the histogram.
There are 3 spaces and 3 maps forming a commutative triangle.
One could say in this case that the mean graylevel "factors through" the histogram.

The same is true if 'mean graylevel' is replaced by other simple image statistics, such as median or any other quantile. Many image APIs, such as ImageJ's macro language, can compute the histogram very quickly. So one can use that factorization to compute any quantile quickly too.

score 1 · Answer 4 · answered Feb 28 '15 at 10:23

Commutative diagrams occur very frequently in category theory, which is very close to type theory in computational science/software engineering, c.f.: https://en.wikipedia.org/wiki/Type_theory#Relation_to_category_theory

How can an engineer make use of commutative diagram?

4 Answers4