To build a matrix of differences between the elements of a vector, you could simply get the vector in column form and substract its transposed (row) vector. Similarly to what you would do in matrix multiplication but by subtracting instead of multiplying:
--------1-------2-------3------4
1---(1-1) (1-2)
2---(2-1) ... et cetera
3---(3-1)
4---(4-1)
Giving as a result the matrix of differences between the elements of the vector.
However, this very simple operation is never defined in basic introductory courses or texts to matrix algebra. Is there a name for this operation, like "transposed vector subtraction" or "col-row vector subtraction"?. Why isn't this very simple operation defined in general matrix algebra?
Moreover, not all software packages allow it. For instance, I can perform it in MatLab simply by coding:
X - X' ;
where X is a column vector and X' is its transpose
(while, if I wanted to get the element-by-element difference, the code should be: X .- X')
...But, if I do the same in R, it simply doesn't do it. I had to create a for-loop to code it.
Hello.
I would say that new operators should be introduced only in such a case if we are not able to reach desired effect by existing ones. Here we need only transposition and subtraction (you already outlined the basic idea). Matrix transposition and subtraction should be done easily in any mathematical engine and there is no need to use cycles.
Please, check the PDF attachment below.
Bye, JS
Related to your query I suggest you read Hilbert-Spaces and Banach Spaces over linear operators. A simple answer to your query, may however be here : http://www.robots.ox.ac.uk/~albanie/notes/Euclidean_distance_trick.pdfhttps://reference.wolfram.com/language/ref/DistanceMatrix.html
Muhammad Ali thanks a lot!
Awesome, what you sent me in the files makes perfect sense... I did not know that answer, thanks a lot.
However saying " new operators should be introduced only if we are not able to reach desired effect by existing ones " ... would mean that, in basic arithmetic, the multiplication and exponentiation operations are not needed because:
4 x 3 = 4 + 4 + 4
and
4^2 = 4+4+4+4
You must agree that your answer is quite complicated. I'm thinking about myself at 11 years old, on my first introductory class to matrix algebra... thinking, what is going on!!!!... Why the first opperation is not a sum??!?!?!
Hello.
I would say, that your question has been addressed to me rather than dr. Muhammad Ali. So let me answer it.
So: " Why the first opperation is not a sum??!?!?! "
Operation 4x3 really can be expressed as 4x3=4+4+4. But how would you manage to evaluate following multiplication using the same logic?
4xPi=4+4+4+?
As for exponentiation let's move from 4^2 to 4^(2.1)=4+4+4+4+? How would you do it (without using infinite Taylor series)?
The real importance of all the elementary algebraic operations gets clear just after moving from integers to rational or real numbers. All the elementary operations mentioned above are essential to cover all the human mathematical needs.
And it is similar with vectors and matrices. Nearly anything we need to evaluate can be done by already defined set of operations.
Your original question was: " Why isn't this very simple operation defined in general matrix algebra?" So I would say, we do not need it :-)
Bye bye, JS
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