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Eigenvectors and Eigenvalues (2015)

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Re: Eigenvectors and Eigenvalues (2015)

#61

Post by xevb3k »

ItsMe000001 wrote: post #46
angersock wrote: post #39

Earlier quoted context omitted.

Because, say, knowing about Fourier transforms can help you write more efficient filtering or open up new ways to view your data--perhaps there's a really interesting behavior in the frequency domain you'd miss otherwise. If you just want to be a statistical script kiddie you do you. :)

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D.…

Really interesting, thanks! Yes, it’s certainly been my experience that adding more domain knowledge has helped me a huge amount.

Re: Eigenvectors and Eigenvalues (2015)

#62

Post by hyperpallium »

ItsMe000001 wrote: post #46
angersock wrote: post #39

Earlier quoted context omitted.

Because, say, knowing about Fourier transforms can help you write more efficient filtering or open up new ways to view your data--perhaps there's a really interesting behavior in the frequency domain you'd miss otherwise. If you just want to be a statistical script kiddie you do you. :)

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D.…

I'm doing some fluid simulation (CFD), and the actual code for finite differences is simple. But the analysis (of stability etc) is more mathy, and I don't feel confident reading other needed papers, because they are couched in math.

I mean, I can and have coded it, but can't be sure how it how it will behave in all situations.

So I can understand these papers, I'm going back to study math properly. I'm not fully convinced it's really needed (though how could I tell?), but I'm fully convinced it's needed to understand the papers.

Math is the latin of CFD.

Re: Eigenvectors and Eigenvalues (2015)

#63

Post by hyperpallium »

pphysch wrote: post #49
ItsMe000001 wrote: post #46

Earlier quoted context omitted.

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D.…

As a younger person (finishing up a Math BS) this resonates with my perspective. IMHO, it comes down to individual beliefs about mathematical realism. Is there anything inherently real about math, or is it just a man-made, arbitrary set of cognitive tools? Is it valid to presume the existence of a Grand Mathematical Framework that can solve any problem a priori? Or, is every problem unique and independent of mathemat…

Reminds me of economic rationalism: model makes sense, and conclusions seem logical and compelling - but you can't tell if the model represents reality or even if crucial factors have been omitted.

Re: Eigenvectors and Eigenvalues (2015)

#64

Post by picklesman »

AceJohnny2 wrote: post #38
steamer25 wrote: post #5

3Blue1Brown has a good series on YouTube for building intuition in linear algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... In one of the last videos in the (relatively short) series, he discusses eigen-*: ~'eigen-stuffs are straight-forward but only make sense if you have a solid visual understanding of the pre-requisites (linear transformations, determinants, linear systems of equations, chan…

3Blue1Brown (Grant Sanderson) is really, really good. I follow a number of education channels on YouTube, and Grant blows them all out of the water for the kind of insights, new perspectives, and inspiration he provides. His animations are fantastically put together to clearly and unobtrusively illustrate the point he's making. I also really like his voice, soothing, clear and with enough intonation to avoid boredom,…

I’ve taken 3 college level courses on linear algebra and only really got an intuition for eigenvectors/eigenvalues when watching his video series. Super great!

Re: Eigenvectors and Eigenvalues (2015)

#65

Post by bigger_cheese »

hyperpallium wrote: post #62
ItsMe000001 wrote: post #46

Earlier quoted context omitted.

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D.…

I'm doing some fluid simulation (CFD), and the actual code for finite differences is simple. But the analysis (of stability etc) is more mathy, and I don't feel confident reading other needed papers, because they are couched in math. I mean, I can and have coded it, but can't be sure how it how it will behave in all situations. So I can understand these papers, I'm going back to study math properly. I'm not fully con…

I work in Engineering and feel somewhat similar about CFD. There's always some element of doubt lurking in the back of my mind "is it really correct in all circumstances".

The most useful info I learnt at university were a couple of equations: Bernoulli's (for general observation about expected pressure drop), Ergun's (for flow through packed beds) and the general laws of thermodynamics.

Those are mostly enough to be able to sketch out an intuitive 'guess' about expected behavior in a large range of systems and the underlying math is not particularly demanding.

Re: Eigenvectors and Eigenvalues (2015)

#66

Post by Retra »

haskellandchill wrote: post #28
pizza wrote: post #24

Earlier quoted context omitted.

Does this gif help? https://commons.wikimedia.org/wiki/File:Eigenvectors.gif

No gif will help. There is no visual explanation that will help was my point.

Surely visual information helps with geometric problems? Our geometric intuition basically evolved to predict the relevance of light patterns on our retinas, and geometry is a language designed to encode these intuitions, so there's no reason to think visual tools presenting geometric facts would inherently fall short.

Re: Eigenvectors and Eigenvalues (2015)

#67

Post by tprice7 »

soVeryTired wrote: post #53
qmalzp wrote: post #48

Earlier quoted context omitted.

I think the first point is only true for symmetric matrices (which includes those that show up in multivariable calc). In general, the eigenvectors need not be orthogonal.

Yep, you could well be right. The image of an ellipse under a linear transform is definitely an ellipse, but I'm not sure about the eigenvectors in the general case. The symmetric case is by far the most relevant for probability theory though.

In general it's the eigenvectors of the positive-semidefinite (hence symmetric) part of the left polar decomposition.

Re: Eigenvectors and Eigenvalues (2015)

#68

Post by smrq »

steamer25 wrote: post #5

3Blue1Brown has a good series on YouTube for building intuition in linear algebra: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x... In one of the last videos in the (relatively short) series, he discusses eigen-*: ~'eigen-stuffs are straight-forward but only make sense if you have a solid visual understanding of the pre-requisites (linear transformations, determinants, linear systems of equations, chan…

Just to throw in my own anecdote, I took linear algebra twice (in high school with no college credit, and in college) and I still couldn't ever remember afterwards what an eigenvector was until I watched that series. Now I'll probably never forget. He is an astonishing educator.

Re: Eigenvectors and Eigenvalues (2015)

#70

Post by ItsMe000001 »

Jack000 wrote: post #60
ItsMe000001 wrote: post #46

Earlier quoted context omitted.

When working with real world data almost everything is more important than being able to use the most abstract methods "to extract the last bit of data". It's often extremely fuzzy to begin with, the collection process to what it represents, for me, while I love math and see it as the "magical language" in a magical world, I find common sense and a certain kind of work ethics go soooo much further than any math Ph.D.…

well you don't need this stuff until you do - some things still need definite, analytical performance guarantees. I'd be pretty nervous riding an airplane that didn't use modern control theory, or going over a bridge that didn't use FEA - or an self-driving car that ran on a raspberry pi instead of a RTOS...

Why do you think in absolutes? I'm tempted to cite a certain Dilbert... but that would make me appear incredibly rude, which I really don't want to be. Of course it is easy to find examples where it's needed, and it is easy to see so too. So? Did I propose at any point "Nobody needs higher math for anything" or something similar? I don't see a need to argue about an argument never made.
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