Need help understanding summation notion

[NicolaiTheDane]

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

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+..+v_i*a_i (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.

 

[Orodruin]

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.

 

[Mark44]

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+..+v_i*a_i (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.

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.

 

[StoneTemplePython]

$\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow \sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j) \Leftrightarrow \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)$

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.

 

[WWGD]

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+..+v_i*a_i (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...

 

[Orodruin]

https://en.wikipedia.org/wiki/Einstein_notation

Nothing rotten in there...

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.

 

[NicolaiTheDane]

$\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow \sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j) \Leftrightarrow \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)$

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

$v = \sum_{i = 1}^n$ v_i*x_i
and
$w = \sum_{i = 1}^n$ v_i*y_i

leads to the following:

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.

 

[StoneTemplePython]

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.

The assignment I needed to do, basically seems to be about quantum mechanics (haven't had anything on this yet), where

$v = \sum_{i = 1}^n$ v_i*x_i
and
$w = \sum_{i = 1}^n$ v_i*y_i

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.

 

[Mark44]

$\sum_{1 \leq i < j \leq n} f(i,j) \Leftrightarrow \sum_{i = 1}^n \sum_{j = i + 1}^n f(i,j) \Leftrightarrow \sum_{i = 1}^{n-1} \sum_{j = i + 1}^n f(i,j)$

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.

 

[StoneTemplePython]

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 ${\LaTeX}$.

 

[NicolaiTheDane]

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!