14 ms·

Eigenvectors and eigenvalues explained visually

523 points by vicapow 12y ago

Terr_ 12y ago
Very cool, but as a layman I was a very confused by the description of eigenspaces and the S1/S2 lines. I'm just guessing here (reasoning below) but I'd like to suggest phrasing like: "Eigenspaces are special lines, where any starting-point along them yields an eigenvalue that lands back on the same line. In these examples two exist, labeled, S1 and S2." "Eigenspaces show where there is 'stability' from repeated applications of the eigenvector. Some act like 'troughs' which attract nearby series of points (S1) while others are like hills (S2) where any point even slightly outside the stable peak yields eigenvalues further away. ______ Original post / detailed-reaction: > First, every point on the same line as an eigenvector is another eigenvector. That line is an eigenspace. At first I though this statement-of-fact meant that the whole tweakable quadrant of the X/Y plot (at a minimum) is an unbroken 2D Eigenspace, because every point within it can be "covered" by a dashed line (a 2D "vector") if I pick the appropriate start-point. However, the last sentence also says eigenspaces are (despite the "space" in their name) lines, which throws the earlier interpretation into doubt. > As you can see below, eigenspaces attract this sequence S1 and S2 were displayed earlier, but not explained, now this section implies that those lines are the Eigenspaces? If so, what is the difference between S1 and S2? Playing with the chart, I assume they are the "forward" and "reverse" for repeat-applications of the transformation.
- lewis500 12y ago
  hey yeah good point. I'll add a reference to the labels, which were added last minute.
TehCorwiz 12y ago
The interactive graph in the section "Complex eigenvalues" has a repeatable crash bug in Chrome 39 on Win 7. There are a number of was to trigger it, the easiest of which is to adjust a1 and a2 such that both have positive x and y values and the resulting line from v to Av has a slope of approximately 1.
bsaul 12y ago
Very beautiful graphs, but i don't think it's going to make people understand anything. I would start with the problem : easily compute sums of dependant values, then show a naive computation, then use matrices, vectors and eigenvalues to come to a solution, and only then, show a graphical representation of the steps performed. I'm surprised that this post isn't following this method, because i've come to think it's the standard way of explaining scientific things in the US.
- floatrock 12y ago
  Eigenvalues and eigenvectors are one of those things that pop up in a million places because they're so useful, but to recognize where they may be useful you need intuition as to what they're doing. One of my biggest hurdles learning linear algebra was getting that intuition. This "standard way of explaining scientific things" never built that intuition for me, only forced the mechanics of the computation into pencilized muscle memory. Don't get me wrong, you need that muscle memory in practice. But without the intuition, your muscle memory is always going to be inferior to a couple commands in matlab. Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot. Play (and simulation) is an extremely effective way to build intuition, and one I'd love to see more. These guys are doing an awesome job at these kinds of simulations -- their markov chain one was fantastic too http://setosa.io/ev/markov-chains/ http://setosa.io/ev/markov-chains/
  - na85 12y ago
    >Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot. Disagree. The linked article does a very poor job of explaining anything, much less conveying intuition. The lambda values remain even if the 3 points are not collinear, thus contradicting the first part of the article.
- ris 12y ago
  "Very beautiful graphs, but i don't think it's going to make people understand anything." Incorrect. This is the first time I've ever really understood eigen*s
michaf 12y ago
I like the visualization. But there seems to be an error: the non-diagonal elements of the Markov matrix need to be interchanged. You can see this by setting p=1 and q=0. Their formula would result in a total population of 2*California after one step, which is clearly larger than California+New York.
- lewis500 12y ago
  good catch my friend
discardorama 12y ago
The pretty animations are nice, and the ability to manipulate the vectors is very nice; however, I am sorry to say (and I do not mean this negatively) that there's not much "explanation". The first sentence just describes the utility of the Eigens (so no explanation there). The next lays out the setting for the diagram. And the third says, "if we can do X, then v is an eigenvector and \lambda an eigenvalue". But... what if you can't do "X" ? What if v, (0,0) and Av are not colinear? The skeleton of a great explanation is there, but the meat isn't there yet. A few more sentences would go a long way in making this better. I appreciate the OP's effort, and I hope this will come across as constructive criticism.
- tptacek 12y ago
  WRITE THOSE SENTENCES! Don't leave us hanging!
  - b_emery 12y ago
    This from wikipedia is where it started to click for me: "In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix". So the eigenvectors are like the directions that describe position of an airplane: roll pitch and yaw.
    - JadeNB 12y ago
      This point of view was used in a recent HN post: https://news.ycombinator.com/item?id=8904089 https://news.ycombinator.com/item?id=8904089 .
  - kalid 12y ago
    My take: The eigenvectors are the “axes” of the transformation represented by the matrix. Consider spinning a globe (the universe of vectors): every location faces a new direction, except the poles. An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix. (Shameless plug, more here: http://betterexplained.com/articles/linear-algebra-guide/ http://betterexplained.com/articles/linear-algebra-guide/)
kristopolous 12y ago
It takes me an enormous amount of effort to read this font. I need to squint my eyes and had to zoom my browser window to about 200% and then scroll horizontally to make my way through the paragraphs.
- apenguin 12y ago
  Try changing the font-width instead. I absolutely hate it when websites use fonts this thin, too.
  - kristopolous 12y ago
    are you talking about through the css? What method do you use? I really want to read this content but I find some difficulty in it.
    - apenguin 12y ago
      Yeah, I changed the CSS with the inspector. I don't know if it's because I have a low resolution screen, but I can't actually read it at all without doing this.
mturmon 12y ago
>> "It turns out that a matrix like A, whose rows add up to zero (try it!), is called a Markov matrix, ..." Oops, you mean the rows add to one. I hate to nitpick, but, additionally, numbers in the matrix can't be negative. Also, it's not just that 1 is an eigenvalue, it is that 1 is the largest eigenvalue. This is significant, because it implies that all other components will die out in time.
- pbhjpbhj 12y ago
  The cited source says that in a Markov matrix the columns sum to 1 which Wikipedia reports is a "left stochastic matrix" whilst the rows summing to 1 makes it a "right stochastic matrix" - I'm not sure on the definition of Markov Matrix specifically but then it doesn't seem that http://mathworld.wolfram.com/StochasticMatrix.html http://mathworld.wolfram.com/StochasticMatrix.html knows either. The term returns no search results at http://www.encyclopediaofmath.com/ http://www.encyclopediaofmath.com/. http://blog.stata.com/2011/03/09/understanding-matrices-intuitively-part-2/ http://blog.stata.com/2011/03/09/understanding-matrices-intu... came up here before and helped me visualise eigen-stuff.
  - mturmon 12y ago
    Whether it's rows vs. columns is just a matter of definition, depending on whether you want to propagate the state by left multiplication or right multiplication. The typical definition (in the context of Markov chains) has the rows summing to one. But how to do it is up to the author of the page. Just be consistent. In my own coursework, that matrix is called a "stochastic matrix", btw, not a "Markov matrix", but again, it's just definitional and not of interest in a simple article like this. My main point is that summing to one is what you want, and summing to zero is crazy.
    - pbhjpbhj 12y ago
      Indeed, I've heard of stochastic matrices - just wasn't sure they were coterminous in definition with "Markov matrix". Thanks for clarifying though.
noelwelsh 12y ago
As a counterpoint to most of the comments, let me just say: this is fantastic. (Nothing wrong with constructive criticism, which most comments are, but it's also nice to just say thanks as well.)
hangonhn 12y ago
The graphics is nice but the explanation is just terrible. There is a huge gap between the first part explaining vectors and then the part explaining eigenvectors. "If you can draw a line through (0,0), v and Av, then Av is just v multiplied by a number λ; that is, Av=λv." That makes no sense. How do you draw a line through a point to "v and Av"? What does "v and Av" even mean in that context?
- Oatmeat 12y ago
  If you can draw a line through the three points: (0,0), v, and Av
  - hangonhn 12y ago
    That assumption should have been stated in the explanation and the graphics need to reflect that constraint, right? Or am I still missing something here?
hangonhn 12y ago
This wikipedia graphic gives a pretty good graphical explanation of what eigenvalues do and what eigenvectors are: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#mediaviewer/File:Eigenvectors.gif http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#me...
acd 12y ago
When I see similar faces that look a like from different people I wonder if they have similar eigenfaces?
- corysama 12y ago
  I'm pretty sure they do: http://en.wikipedia.org/wiki/Eigenface http://en.wikipedia.org/wiki/Eigenface
- spacemanmatt 12y ago
  I would not be surprised if FB had a class called Eigenperson, somewhere in their code or database.
  - deleted 12y ago
    [deleted]
throw7 12y ago
I have no idea what eigenvectors or eigenvalues are, so this just confused me more. To be fair, I think the author does assume some basic math understanding before hand though.
debacle 12y ago
I have to admit I hated the term Eigenvector for two semesters of college and it nearly caused me to drop mathematics altogether. This explanation is very good and helps visualize some of the things I was missing. Apologies to the fantastic professors I had who were talking over my head for 16 weeks.
dlwj 12y ago
First time I've actually sort of understood eigenvectors. Linear algebra was actually the class that made me hate math, after years of loving it in secondary education. Not everyone has the benefit of a good teacher, and the tools that exist now don't help you to self-learn much.
xixixao 12y ago
Wow, almost no positive feedback here? I think the article assumes certain audience, and for me, this brought a great insight I did not ever get in our college courses.
pcvarmint 12y ago
Did you send this to Malcolm Gladwell? :) Igon send it to him if you can't :)
cousin_it 12y ago
I just tried to figure out the simplest rigorous explanation of linear transformations. Here's one in terms of straight lines. Let's say we have a transformation of the 2D plane, i.e. a mapping from points to points. We will call that a "linear transformation" if these conditions are satisfied: 1) The point (0, 0) gets mapped to itself. 2) Straight lines get mapped to straight lines, though maybe pointing in a different direction. 3) Pairs of parallel straight lines get mapped to pairs of parallel straight lines. Hence the name "linear transformation" :-) We can see that all straight lines going through (0, 0) get mapped to straight lines going through (0, 0). Let's consider just those straight lines going through (0, 0) that get mapped to themselves. There are four possibilities: 1) There are no such lines, e.g. if the transformation is a rotation. 2) There is one such line, e.g. if the transformation is a skew. 3) There are two such lines, e.g. if the transformation is a stretch along some axis. 4) There are more than two such lines. In this case, you can prove that in fact all straight lines going through (0, 0) are mapped to themselves, and the transformation is a scaling. Now let's consider what happens within a single such line that gets mapped to itself. You can prove that within a single such line, the transformation becomes a scaling by some constant factor. (That factor could also be negative, which corresponds to flipping the direction of the line.) Let's call these factors the "eigenvalues", or "own values" of the transformation. Now let's define the "eigenspaces", or "own spaces" of the transformation, corresponding to each eigenvalue. An eigenspace is the set of all points in the 2D plane for which the transformation becomes scaling by an eigenvalue. Let's see what happens in each of the cases: 1) In case 1, there are no eigenspaces and no eigenvalues. 2) In case 2, there is only one eigenspace, which is the straight line corresponding to the single eigenvalue. 3) In case 3, it pays off to be careful! First we need to check what happens if the two eigenvalues are equal. If that happens, it's easy to prove that we end up in case 4 instead. Otherwise there are two different eigenvalues, and their eigenspaces are two different straight lines. 4) In case 4, the eigenspace is the whole 2D plane. In this way, eigenvalues and eigenspaces are unambiguously geometrically defined, and don't require coordinates or matrices. Now, what are "eigenvectors", or "own vectors" of the transformation? Let's say that an "eigenvector" is any vector for which our transformation is a scaling. In other words, an "eigenvector" is a vector from (0, 0) to any point in an eigenspace. The disadvantage is that it involves an arbitrary choice. The advantage is that eigenvectors can be specified by coordinates, so you can find them by computational methods. Does that make sense?

523 points by vicapow 12y ago

Terr_ 12y ago

Very cool, but as a layman I was a very confused by the description of eigenspaces and the S1/S2 lines. I'm just guessing here (reasoning below) but I'd like to suggest phrasing like: "Eigenspaces are special lines, where any starting-point along them yields an eigenvalue that lands back on the same line. In these examples two exist, labeled, S1 and S2." "Eigenspaces show where there is 'stability' from repeated applications of the eigenvector. Some act like 'troughs' which attract nearby series of points (S1) while others are like hills (S2) where any point even slightly outside the stable peak yields eigenvalues further away. ______ Original post / detailed-reaction: > First, every point on the same line as an eigenvector is another eigenvector. That line is an eigenspace. At first I though this statement-of-fact meant that the whole tweakable quadrant of the X/Y plot (at a minimum) is an unbroken 2D Eigenspace, because every point within it can be "covered" by a dashed line (a 2D "vector") if I pick the appropriate start-point. However, the last sentence also says eigenspaces are (despite the "space" in their name) lines, which throws the earlier interpretation into doubt. > As you can see below, eigenspaces attract this sequence S1 and S2 were displayed earlier, but not explained, now this section implies that those lines are the Eigenspaces? If so, what is the difference between S1 and S2? Playing with the chart, I assume they are the "forward" and "reverse" for repeat-applications of the transformation.

lewis500 12y ago
hey yeah good point. I'll add a reference to the labels, which were added last minute.

TehCorwiz 12y ago

The interactive graph in the section "Complex eigenvalues" has a repeatable crash bug in Chrome 39 on Win 7. There are a number of was to trigger it, the easiest of which is to adjust a1 and a2 such that both have positive x and y values and the resulting line from v to Av has a slope of approximately 1.

bsaul 12y ago

Very beautiful graphs, but i don't think it's going to make people understand anything. I would start with the problem : easily compute sums of dependant values, then show a naive computation, then use matrices, vectors and eigenvalues to come to a solution, and only then, show a graphical representation of the steps performed. I'm surprised that this post isn't following this method, because i've come to think it's the standard way of explaining scientific things in the US.

floatrock 12y ago
Eigenvalues and eigenvectors are one of those things that pop up in a million places because they're so useful, but to recognize where they may be useful you need intuition as to what they're doing. One of my biggest hurdles learning linear algebra was getting that intuition. This "standard way of explaining scientific things" never built that intuition for me, only forced the mechanics of the computation into pencilized muscle memory. Don't get me wrong, you need that muscle memory in practice. But without the intuition, your muscle memory is always going to be inferior to a couple commands in matlab. Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot. Play (and simulation) is an extremely effective way to build intuition, and one I'd love to see more. These guys are doing an awesome job at these kinds of simulations -- their markov chain one was fantastic too http://setosa.io/ev/markov-chains/ http://setosa.io/ev/markov-chains/
- na85 12y ago
  >Stuff like this visualization builds the intuitive knowledge -- you see, you explore, you wiggle a few things and see what happens when you go in and out of the sweet spot. Disagree. The linked article does a very poor job of explaining anything, much less conveying intuition. The lambda values remain even if the 3 points are not collinear, thus contradicting the first part of the article.
ris 12y ago
"Very beautiful graphs, but i don't think it's going to make people understand anything." Incorrect. This is the first time I've ever really understood eigen*s

michaf 12y ago

I like the visualization. But there seems to be an error: the non-diagonal elements of the Markov matrix need to be interchanged. You can see this by setting p=1 and q=0. Their formula would result in a total population of 2*California after one step, which is clearly larger than California+New York.

lewis500 12y ago
good catch my friend

discardorama 12y ago

The pretty animations are nice, and the ability to manipulate the vectors is very nice; however, I am sorry to say (and I do not mean this negatively) that there's not much "explanation". The first sentence just describes the utility of the Eigens (so no explanation there). The next lays out the setting for the diagram. And the third says, "if we can do X, then v is an eigenvector and \lambda an eigenvalue". But... what if you can't do "X" ? What if v, (0,0) and Av are not colinear? The skeleton of a great explanation is there, but the meat isn't there yet. A few more sentences would go a long way in making this better. I appreciate the OP's effort, and I hope this will come across as constructive criticism.

tptacek 12y ago
WRITE THOSE SENTENCES! Don't leave us hanging!
- b_emery 12y ago
  This from wikipedia is where it started to click for me: "In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix". So the eigenvectors are like the directions that describe position of an airplane: roll pitch and yaw.
  - JadeNB 12y ago
    This point of view was used in a recent HN post: https://news.ycombinator.com/item?id=8904089 https://news.ycombinator.com/item?id=8904089 .
- kalid 12y ago
  My take: The eigenvectors are the “axes” of the transformation represented by the matrix. Consider spinning a globe (the universe of vectors): every location faces a new direction, except the poles. An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix. (Shameless plug, more here: http://betterexplained.com/articles/linear-algebra-guide/ http://betterexplained.com/articles/linear-algebra-guide/)

kristopolous 12y ago

It takes me an enormous amount of effort to read this font. I need to squint my eyes and had to zoom my browser window to about 200% and then scroll horizontally to make my way through the paragraphs.

apenguin 12y ago
Try changing the font-width instead. I absolutely hate it when websites use fonts this thin, too.
- kristopolous 12y ago
  are you talking about through the css? What method do you use? I really want to read this content but I find some difficulty in it.
  - apenguin 12y ago
    Yeah, I changed the CSS with the inspector. I don't know if it's because I have a low resolution screen, but I can't actually read it at all without doing this.

mturmon 12y ago

>> "It turns out that a matrix like A, whose rows add up to zero (try it!), is called a Markov matrix, ..." Oops, you mean the rows add to one. I hate to nitpick, but, additionally, numbers in the matrix can't be negative. Also, it's not just that 1 is an eigenvalue, it is that 1 is the largest eigenvalue. This is significant, because it implies that all other components will die out in time.

pbhjpbhj 12y ago
The cited source says that in a Markov matrix the columns sum to 1 which Wikipedia reports is a "left stochastic matrix" whilst the rows summing to 1 makes it a "right stochastic matrix" - I'm not sure on the definition of Markov Matrix specifically but then it doesn't seem that http://mathworld.wolfram.com/StochasticMatrix.html http://mathworld.wolfram.com/StochasticMatrix.html knows either. The term returns no search results at http://www.encyclopediaofmath.com/ http://www.encyclopediaofmath.com/. http://blog.stata.com/2011/03/09/understanding-matrices-intuitively-part-2/ http://blog.stata.com/2011/03/09/understanding-matrices-intu... came up here before and helped me visualise eigen-stuff.
- mturmon 12y ago
  Whether it's rows vs. columns is just a matter of definition, depending on whether you want to propagate the state by left multiplication or right multiplication. The typical definition (in the context of Markov chains) has the rows summing to one. But how to do it is up to the author of the page. Just be consistent. In my own coursework, that matrix is called a "stochastic matrix", btw, not a "Markov matrix", but again, it's just definitional and not of interest in a simple article like this. My main point is that summing to one is what you want, and summing to zero is crazy.
  - pbhjpbhj 12y ago
    Indeed, I've heard of stochastic matrices - just wasn't sure they were coterminous in definition with "Markov matrix". Thanks for clarifying though.

noelwelsh 12y ago

As a counterpoint to most of the comments, let me just say: this is fantastic. (Nothing wrong with constructive criticism, which most comments are, but it's also nice to just say thanks as well.)

hangonhn 12y ago

The graphics is nice but the explanation is just terrible. There is a huge gap between the first part explaining vectors and then the part explaining eigenvectors. "If you can draw a line through (0,0), v and Av, then Av is just v multiplied by a number λ; that is, Av=λv." That makes no sense. How do you draw a line through a point to "v and Av"? What does "v and Av" even mean in that context?

Oatmeat 12y ago
If you can draw a line through the three points: (0,0), v, and Av
- hangonhn 12y ago
  That assumption should have been stated in the explanation and the graphics need to reflect that constraint, right? Or am I still missing something here?

hangonhn 12y ago

This wikipedia graphic gives a pretty good graphical explanation of what eigenvalues do and what eigenvectors are: http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#mediaviewer/File:Eigenvectors.gif http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors#me...

acd 12y ago

When I see similar faces that look a like from different people I wonder if they have similar eigenfaces?

corysama 12y ago
I'm pretty sure they do: http://en.wikipedia.org/wiki/Eigenface http://en.wikipedia.org/wiki/Eigenface
spacemanmatt 12y ago
I would not be surprised if FB had a class called Eigenperson, somewhere in their code or database.
- deleted 12y ago
  [deleted]

throw7 12y ago

I have no idea what eigenvectors or eigenvalues are, so this just confused me more. To be fair, I think the author does assume some basic math understanding before hand though.

debacle 12y ago

I have to admit I hated the term Eigenvector for two semesters of college and it nearly caused me to drop mathematics altogether. This explanation is very good and helps visualize some of the things I was missing. Apologies to the fantastic professors I had who were talking over my head for 16 weeks.

dlwj 12y ago

First time I've actually sort of understood eigenvectors. Linear algebra was actually the class that made me hate math, after years of loving it in secondary education. Not everyone has the benefit of a good teacher, and the tools that exist now don't help you to self-learn much.

xixixao 12y ago

Wow, almost no positive feedback here? I think the article assumes certain audience, and for me, this brought a great insight I did not ever get in our college courses.

pcvarmint 12y ago

Did you send this to Malcolm Gladwell? :) Igon send it to him if you can't :)

cousin_it 12y ago

I just tried to figure out the simplest rigorous explanation of linear transformations. Here's one in terms of straight lines. Let's say we have a transformation of the 2D plane, i.e. a mapping from points to points. We will call that a "linear transformation" if these conditions are satisfied: 1) The point (0, 0) gets mapped to itself. 2) Straight lines get mapped to straight lines, though maybe pointing in a different direction. 3) Pairs of parallel straight lines get mapped to pairs of parallel straight lines. Hence the name "linear transformation" :-) We can see that all straight lines going through (0, 0) get mapped to straight lines going through (0, 0). Let's consider just those straight lines going through (0, 0) that get mapped to themselves. There are four possibilities: 1) There are no such lines, e.g. if the transformation is a rotation. 2) There is one such line, e.g. if the transformation is a skew. 3) There are two such lines, e.g. if the transformation is a stretch along some axis. 4) There are more than two such lines. In this case, you can prove that in fact all straight lines going through (0, 0) are mapped to themselves, and the transformation is a scaling. Now let's consider what happens within a single such line that gets mapped to itself. You can prove that within a single such line, the transformation becomes a scaling by some constant factor. (That factor could also be negative, which corresponds to flipping the direction of the line.) Let's call these factors the "eigenvalues", or "own values" of the transformation. Now let's define the "eigenspaces", or "own spaces" of the transformation, corresponding to each eigenvalue. An eigenspace is the set of all points in the 2D plane for which the transformation becomes scaling by an eigenvalue. Let's see what happens in each of the cases: 1) In case 1, there are no eigenspaces and no eigenvalues. 2) In case 2, there is only one eigenspace, which is the straight line corresponding to the single eigenvalue. 3) In case 3, it pays off to be careful! First we need to check what happens if the two eigenvalues are equal. If that happens, it's easy to prove that we end up in case 4 instead. Otherwise there are two different eigenvalues, and their eigenspaces are two different straight lines. 4) In case 4, the eigenspace is the whole 2D plane. In this way, eigenvalues and eigenspaces are unambiguously geometrically defined, and don't require coordinates or matrices. Now, what are "eigenvectors", or "own vectors" of the transformation? Let's say that an "eigenvector" is any vector for which our transformation is a scaling. In other words, an "eigenvector" is a vector from (0, 0) to any point in an eigenspace. The disadvantage is that it involves an arbitrary choice. The advantage is that eigenvectors can be specified by coordinates, so you can find them by computational methods. Does that make sense?