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Can someone explain to me what is the difference between separable and non-separable function? I found some explanations, but these explanations are not in "human form". For example, I found that separable function can be expressed like this:

g(x,y) = gx(x)gy(y).

It is fine, but can someone explains it in "human form"? I mean some easy sentence, which describes what is a separable function, without some mathematics notation.

Thank you very much.

Atinesh
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Petr
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    you can split the function of two variables into a product of a function of one variable and a function of the other variable e.g. $x\sin(y)$ is a product of $f(x) = x$ and $g(y) = \sin(y)$. On the other hand, $\sin(xy)$ is not separable. – Paul Mar 12 '15 at 11:32
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    Ok, thanks for your explanation, Paul. So, for example, if I have a simple function 10cos(x)5sin(y), this function is separable, because it is a product of f(x) = 10cos(x) and g(y) = 5sin(y). Am I correct? And the second question: Can be some function, which has more than 2 variables, also separable? – Petr Mar 12 '15 at 11:37
  • Yes that is right – Paul Mar 12 '15 at 11:39
  • Ok, thank you very much. Last question, can I some something like this? The function, which has n variables, is separable in case, that this function can be expressed as a product of n functions which have one variable. – Petr Mar 12 '15 at 11:48
  • That would make sense with your definition of separable. As a formula $f(x_1, x_2, ...,x_n) = f_1(x_1)f_2(x_2)...f_n(x_n)$ – Paul Mar 12 '15 at 11:52
  • Ok, I understand. In other words, your formula above is same as my sentence above. That is exactly what I need - thank you very much! – Petr Mar 12 '15 at 11:54
  • Please clarify the meaning of "human form" – Atinesh May 22 '17 at 08:37

1 Answers1

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My suggestion for a "human form" explanation is that if you were plot a separable function f(x,y) you could just look down the x-axis and see a "one-dimensional" function g(x) (technically g(x) = f(x,0)). Any other one-dimensional function parallel to this one (parallel to the x-axis) would be a vertically-scaled version of this function.

It is the "continuous" version of a matrix formed by an outer product, so perhaps the following example gives the full gist of it. We "should" be calling matrices like this one "separable" matrices:

$$\begin{bmatrix} 1 & 2 & 4 & 7 & 2\\ 2 & 4 & 8 & 14 & 4\\ 5 & 10 & 20 & 35 & 10 \end{bmatrix}$$

Note that each column is a multiple of any other column, and the same is true of the rows. The graph of a separable function would have the same property.

It should be clear now that knowing just one row and one column is enough to defined the whole matrix. This fact answers the last question: yes, it is true that a separable function (in N-variables) can be defined using just N 1-variable functions.

Asaf Karagila
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user340793
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