Need help understanding summation notion

  • Thread starter NicolaiTheDane
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In summary, the notation ##\sum_{1 \leq i < j \leq n} f(i,j)## represents a sum over all combinations of integer ##i## and ##j## such that ##i < j## and both ##i## and ##j## are between 1 and n. This can also be shown as two separate sums, where the first index ranges from 1 to n and the second index ranges from the first index plus one to n. This notation is commonly used in linear algebra and can also be seen as a sum over the upper triangular cells in an associated matrix.
  • #1
NicolaiTheDane
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Homework Statement


Through out my linear algebra book, this weird summation sign has started appearing, and I haven't been able to find anything on it online. Can someone please explain how I'm suppose to read this:

Homework Equations


upload_2017-12-18_22-14-43.png

upload_2017-12-18_22-15-6.png


The Attempt at a Solution


Now in the first case, I could at least work with it, as I could read that notion "i in I" means a family of vectors/scalars, where "I" denotes the entire family, and i denotes an individual vector/scalar. So basically what this means is:
v1*a1+v2*a2+..+vi*ai (where i denotes the last elements)

Now the problem is that the second one, I cannot just read an example and "just do it" as that one from an actually assignment. I need to understand what it actually means. My teacher has barely any time to explain things, as he has his own stuff to get through, but basically from what I can gather, it a sum within a sum. However exactly how this is suppose to work, from the given notation, I haven't got the faintest idea.

Thanks in advance for any all assistance.
 

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  • #2
What is written underneath the summation sign is a description of the set to be summed over. In the case
$$
\sum_{1 \leq i < j \leq n}
$$
this set is all combinations of integer ##i## and ##j## such that ##i < j## and both ##i## and ##j## are between 1 and n.
 
  • #3
NicolaiTheDane said:

Homework Statement


Through out my linear algebra book, this weird summation sign has started appearing, and I haven't been able to find anything on it online. Can someone please explain how I'm suppose to read this:

Homework Equations


View attachment 217024
View attachment 217025

The Attempt at a Solution


Now in the first case, I could at least work with it, as I could read that notion "i in I" means a family of vectors/scalars, where "I" denotes the entire family, and i denotes an individual vector/scalar. So basically what this means is:
v1*a1+v2*a2+..+vi*ai (where i denotes the last elements)
The I (capital letter I) possibly means integers, or positive integers. You might find an explanation of this notation in the front of your book or in the back.
NicolaiTheDane said:
Now the problem is that the second one, I cannot just read an example and "just do it" as that one from an actually assignment. I need to understand what it actually means. My teacher has barely any time to explain things, as he has his own stuff to get through, but basically from what I can gather, it a sum within a sum. However exactly how this is suppose to work, from the given notation, I haven't got the faintest idea.
This notation means that both indexes range from 1 through n.
 
  • #4
##
\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow
\sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j)
\Leftrightarrow
\sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)
##

if you drew out the associated matrix-- you'd see it's referencing the upper half of it (i.e. strictly upper triangular cells in the array).

I sometimes get annoyed when they show it as a single sigma even though its really a double sum. Such notation is common though.
 
  • #5
NicolaiTheDane said:

Homework Statement


Through out my linear algebra book, this weird summation sign has started appearing, and I haven't been able to find anything on it online. Can someone please explain how I'm suppose to read this:

Homework Equations


View attachment 217024
View attachment 217025

The Attempt at a Solution


Now in the first case, I could at least work with it, as I could read that notion "i in I" means a family of vectors/scalars, where "I" denotes the entire family, and i denotes an individual vector/scalar. So basically what this means is:
v1*a1+v2*a2+..+vi*ai (where i denotes the last elements)

Now the problem is that the second one, I cannot just read an example and "just do it" as that one from an actually assignment. I need to understand what it actually means. My teacher has barely any time to explain things, as he has his own stuff to get through, but basically from what I can gather, it a sum within a sum. However exactly how this is suppose to work, from the given notation, I haven't got the faintest idea.

Thanks in advance for any all assistance.
https://en.wikipedia.org/wiki/Einstein_notation

Nothing rotten in there...
 
  • #6
WWGD said:
Note that this question is about regular summation notation using ##\sum## and the description of the set to sum over written underneath it, not the Einstein summation convention. The Einstein summation convention is a further step in notation.
 
  • #7
StoneTemplePython said:
##
\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow
\sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j)
\Leftrightarrow
\sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)
##

if you drew out the associated matrix-- you'd see it's referencing the upper half of it (i.e. strictly upper triangular cells in the array).

I sometimes get annoyed when they show it as a single sigma even though its really a double sum. Such notation is common though.

How you wrote the last bit, is how I imagined it. I don't see how to comes a upper triangular matrix. As I see it, every time "i" goes up by 1, you have summed f(i,j) through all the j's. A for loop within a for loop. However how you manipulate the sums to get there is my problem. For example I don't see how the first, leads to the middle step, much less how you go from the middle step to the last step. The assignment I needed to do, basically seems to be about quantum mechanics (haven't had anything on this yet), where

##v = \sum_{i = 1}^n## vi*xi
and
##w = \sum_{i = 1}^n## vi*yi

leads to the following:

upload_2017-12-19_10-15-57.png


Like I said, we haven't been told what this means when it comes to the physics, so while I'm sure that's going to be important, its not the focus. We have merely gotten some math rules, and our task is to go from the two sums to the double sum below.
 

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  • #8
NicolaiTheDane said:
How you wrote the last bit, is how I imagined it. I don't see how to comes a upper triangular matrix. As I see it, every time "i" goes up by 1, you have summed f(i,j) through all the j's. A for loop within a for loop. However how you manipulate the sums to get there is my problem. For example I don't see how the first, leads to the middle step, much less how you go from the middle step to the last step.

Have you tried working a specific example here? IF you can't see the upper triangular structure after a simple example, I don't know when you will.

example suppose ##n = 3##. In (i,j) ordered pairs, you get

## \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j) = \sum_{i = 1}^{2} \sum_{j = i + 1}^3 f(i,j) = f(1,2) + f(1,3) + f(2,3)##

the tuples ##\{(1,2), (1,3), (2,3)\}## are the strictly upper triangular cell components of a 3x3 matrix. Nothing more, and nothing less.

Your reference to nested for loops is spot on though. If I ever get confused with summation or product notation I just write it out as for loops in Python and convert back to math-ese. How strong is your programming? If you are good at manipulating for loops then, then basic series manipulations can be built off that.
NicolaiTheDane said:
The assignment I needed to do, basically seems to be about quantum mechanics (haven't had anything on this yet), where

##v = \sum_{i = 1}^n## vi*xi
and
##w = \sum_{i = 1}^n## vi*yi

leads to the following:

View attachment 217054

Like I said, we haven't been told what this means when it comes to the physics, so while I'm sure that's going to be important, its not the focus. We have merely gotten some math rules, and our task is to go from the two sums to the double sum below.

Is this using Winitzki's book? Otherwise, I'm not sure why I'm seeing wedge products here. There's a bit more going on here than just sigma manipulation. For your specific example, I'd suggest fully working through it in a simple case, like when n = 3.
 
  • #9
StoneTemplePython said:
##
\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow
\sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j)
\Leftrightarrow
\sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)
##
Minor nit: The connector symbols above should be =, not ##\Leftrightarrow##. These expressions are equal, not equivalent, which should be reserved for statements that have the same truth value.
 
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  • #10
Mark44 said:
Minor nit: The connector symbols above should be =, not ##\Leftrightarrow##. These expressions are equal, not equivalent, which should be reserved for statements that have the same truth value.

Interesting... I was thinking of them as equivalent representations of the same summation process, but I've never spent much time thinking about this. Serves me right for trying spice up my ##{\LaTeX}##.
 
  • #11
StoneTemplePython said:
Have you tried working a specific example here? IF you can't see the upper triangular structure after a simple example, I don't know when you will.

example suppose ##n = 3##. In (i,j) ordered pairs, you get

## \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j) = \sum_{i = 1}^{2} \sum_{j = i + 1}^3 f(i,j) = f(1,2) + f(1,3) + f(2,3)##

the tuples ##\{(1,2), (1,3), (2,3)\}## are the strictly upper triangular cell components of a 3x3 matrix. Nothing more, and nothing less.

Your reference to nested for loops is spot on though. If I ever get confused with summation or product notation I just write it out as for loops in Python and convert back to math-ese. How strong is your programming? If you are good at manipulating for loops then, then basic series manipulations can be built off that.

Is this using Winitzki's book? Otherwise, I'm not sure why I'm seeing wedge products here. There's a bit more going on here than just sigma manipulation. For your specific example, I'd suggest fully working through it in a simple case, like when n = 3.

I think my problem, is that I have worked with matrices as functions. This might sound silly, but most if the linear algebra I learned prior to attended the Niels Bohr Institute, was from a engineering university, where the focus was much more on "getting dang done". For some reason I just find it hard to think of them in these terms.

That aside, if both upper triangular, you mean above the diagonal, then I see your point. So that the above sum is basically the sum of the elements above the diagonal of a matrix. I think the only way for me to get decent at this, is to slave away on a lot exercises, assuming I can find more then the one example we had in my assignment. If I have understood it wrong, do let me know, but other thanks a lot!
 

Related to Need help understanding summation notion

What is the summation notion?

The summation notion, also known as sigma notation, is a mathematical notation used to represent the sum of a series of numbers or terms. It is denoted by the Greek letter sigma (Σ) and has an upper and lower limit, with the term or expression inside the summation symbol being repeated and added for each value of the variable in the specified range.

What is the purpose of using summation notion?

The purpose of using summation notion is to simplify the representation of a large sum or series of numbers or terms. It allows for complicated sums to be written in a concise and efficient way, making it easier to work with and manipulate mathematically.

How do I read and interpret summation notation?

To read and interpret summation notation, you first identify the variable of the summation (usually indicated by the letter below the sigma symbol), then determine the lower and upper limits of the summation (indicated by the numbers below and above the sigma symbol, respectively). Finally, you evaluate the expression or term inside the sigma symbol for each value of the variable within the specified range and add them together.

What are some common examples of summation notation?

Some common examples of summation notation include finding the sum of the first n natural numbers (Σn = 1+2+3+...+n), finding the sum of a geometric series (Σar^n = a + ar + ar^2 + ... + ar^n), and finding the sum of a series involving factorials (Σn! = 1! + 2! + 3! + ... + n!).

How do I use summation notation in real-world applications?

Summation notation can be used in various real-world applications, such as calculating the total cost of items in a shopping cart (Σcost of each item), finding the average temperature over a period of time (Σtemperature readings / number of readings), and estimating the total distance traveled by a car whose speed changes over time (Σdistance traveled at each speed).

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