What makes Eigenvalues and Eigenvectors important and how were they developed?

[dduardo]

I'm currently taking linear algebra and it has to be the worst math class EVER. It is extremely easy, but I find the lack of application discouraging. I really want to understand how the concepts arose and not simple memorize an algorithm to solve mindless operations, which are tedious. My professor is unhelpful and brushes off any discussion in class citing the lack of time to learn all the material. He also assumes to much, leaving little room for proofs.

If anyone would be kind enough to post the importance of Eigenvalues and Eigenvectors, how they where developed, and possible applications for their use.

Any input is wecome.

 

[dduardo]

Wow, great summary Ambitwistor. Not to metion very clear! You explain things better than Strang himself. I use his book in class -"Introduction to Linear Algebra" and I've listen to his lectures on mit opencourseware but your explanations are much better. Can you teach my class Ambitwistor.

I had no idea that a basis was a set axis. Now it makes sense that the number of vectors in the basis define the dimesion of the vector space.

 

[HallsofIvy]

I'll add a different major applications: differential equations.

The basic theory of linear differential equations IS linear algebra: the set of all solutions to a linear homogeneous differential equations forms a vector space. And 90% of solving non-linear differential equations consists of reducing them to linear equations!

Linear algebra really is the theory of "linear problems".

 

[robinyau]

newbie here

whats the sgnificance of eigenvectors in terms of decribing oscillation?

 

[Dan_potato]

hi I'm a newie also but speaking of eigenvectors what's the importance of normalizing eigenvectors?

Dan

 

[bracey]

Hi I am a newie as well and also very confused by normalising
ok, so now i understand the importance of normalising eigenvectors, but how do you actually do this operation.
for example given the matrix

3 0 0
5 4 0
3 6 1

i calculate the eigenvalues to be 3, 4 and 1
using eigenvalue = 3
i get an eigenvector of

k
-13.5k
-5k

so how do you normalise this, any help would be grately appreciated.

 

[bracey]

brilliant cheers, you were right i typed the eigenvector in wrong

so if i got this right, then the normalsised eigenvector should be

0.069
-0.346
-0.935

 

[bracey]

thanks for the help Ambitwistor, much appreciated

 

[roy]

This is probably going to make me sound really stupid, but as this is my first post go easy on me!

Im abit stuck on working out eigenvectors for 3 by 3 matrices, i get confused when trying to get the realtionships between the three values. Hope that makes sense!

 

[roy]

yeah sorry its not that clear is it, i meant the three terms that make up the eigenvector, i can get eigenvalues alright. take for example the matrix that bracy posted how did you get the
1
-5
-13.5

i can get the equations out of the matrix ok but then get stuck

 

[roy]

ok, let me see if i have got this right, take the same matrix again but this time with an eigenvalue of 4

so you get [3x,5x+4y,3x+6y+z] again but this time it has to equal to 4

3x = 4x
5x+4y = 4y
3x+6y+z = 4z

so the first gives x=1.3 the second gives y=0 and the third gives z=1.3 again

Have i got that right or have i just made an idiot of myself! if i have then i blame the hard day i have had!

 

[roy]

oh yeah, sorry about that, didnt think it looked right!. like i said its been a long hard day!
thank you for the help i will try a few more make sure i got it.

 

[Claire84]

We've just been doing eigenvectors etc and I'm grasping the concepts pretty much ok, it's just my main problem half the time is workin out the eigenvalues which isn't too good since that's usually how you start the question. Take the following matrix A-

-3 7 -5
2 4 3
1 2 2

I just have no idea about how to get them through the normal way of taking the determinant of A-lambdaI equal to zero. Any help would be much appreciated!

My next problem is that when I'm looking at the eigenspace and then doing x+y+z=0 etc to work out what x, y and z need to equal to form an eigenvector, what happens if all 3 eqts are the same? Here's what I have-

6x-2y-4z
3x-y-2z
6x-2y-4z

all equal to zero. Do I take x,y and z as being zero or what? So confused!

 

[selfAdjoint]

Take your matrix
-3 7 -5
2 4 3
1 2 2

subtract λ from each of the numbers on the main diagonal. That is the same as subtracting λI from your matrix.

Now you have
(-3-λ) 7 -5
2 (4-λ) 3
1 2 (2-λ)

Now go through the steps for computing the determinant of this matrix, keeping the lambda factors in parenthesis. You will get a numeric expression with the lambda factors in it. Multiply the factors out and collect terms on powers of lambda. You now have a polynomial in lambda. Set it equal to zero and solve the equation.

 

[Claire84]

Thanks for your help, but as it turned out (after asking one of the pure maths tutors today), the lecturer must have been on the booze when he worte out the examples because most of them can't actually be solved (well not at the level we're at anyway). Typical.

 

[Njorl]

I took linear algebra about 20 years ago. It is comforting to know that nothing has changed. My instructor also stripped the subject of any conceivable applications. The class was boring drugdery, until the entire class fell asleep, then the one interesting nugget of theory was slipped in unnoticed.

If I ever had to teach it, I would dress in a different outlandish costume each day - lederhosen one day, matador outfit the next, then perhaps nothing but a loin cloth. That might keep the students awake.

Oddly enough, it is very important to know if you go into physics or any field with extensive mathematical modelling.

Njorl