this matrix is GOLD

this matrix is GOLD

What does this matrix have to do with the golden ratio? Watch this video and find out! We find the eigenvalues of an interesting 4x4 matrix and see that its proper values are related to the golden ratio. We also mention the eigenvectors and give a cool application to shearing and swapping transformations and sort of steady state Markov chains. This is a must-see for any linear algebra lover, enjoy! This is also probably useful for artificial intelligence and machine learning. The golden ratio was used in renaissance italy to measure the beauty of a sculpture.

[mertaliyigit]

Apparently the nxn case of this (path graph) matrix has eigenvalues 2cos(pi*k/(n+1)) for 1<k<n, how cool is that!

[benburdick]

Matrices like these show up all the time when you're dealing with coupled oscillators in physics! I would have expected the 5x5 case to give the golden ratio though, not the 4x4. Fun stuff!

[TaiserBinJafor]

Your method of matrix diagonalization is so cool.

[pietergeerkens]

Don't you have the second and fourth signs wrong at 6:31 (for the RHS)? The product of the LHS of the first two gives -1 rather than 1, and likewise for the third and fourth.

[RealEverythingComputers]

Love the HDR - a great touch to your video

[akirakato]

when i saw the thumbnail i thought it was some sort of kernel for corner detection or somethig like that

[leofun]

Beautiful. There should exist couple linear transformations with similar properties but all of them can be reduced to this example by permutations.

[Chariotuber]

I totally didn't expect that, so cool!

[rikhalder]

I have a question Sir, how we verify a function is uniform continuous or not by its graph?

[decare]

you swapped 1/φ and -1/φ. This is easily seen by multiplying (1+√5)(1-√5) = -4.

Also, you may want to cut your video and audio at different points so the video can be cut directly after the second snap while the audio is cut a bit later to get the reverb in.

[MrAlvaro]

just at : 6:13 1/phi = -(1-sqrt(5))/2

[ahsanhabibkhan]

@4:25 Dr Peyam, could you explain the reason as to why we can assume the two possibilities?
I always thought you could do that if the RHS has a zero. What if my equation is as follows:
X(x-2) =1 .

The trick does work for this equation, because if you apply it the answers are wrong .
x=1 and (x-2)=1 ???

[davidmurphy]

Sorry, why did you say Mohamed Ali when you were calculating the eigenvector/value? Naming a boxer seems a bit random.

Anyway, I'm going to hit play and see what's so special about this 4d matrix.

[marcelob.]

Wonderful! I think #Colapinto can use this to win tomorrow's grand prix.

[Siroitin]

sir, a matrix is an abstract object so it can't be gold

[abdonecbishop]

6:55: ....recall classical physics continuum mechanics coordinate transform equation....that use the 3D coordinate tensor transform equation... σ′=Q⋅σ⋅Q^T....now make following P = Q, P^-1 = Q^T, and D = σ in the matrix equation A = P ⋅D ⋅ P^1 much more gold and cooler is that?