COMPLEX Eigenvalues, Eigenvectors & Diagonalization **full example**
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- Опубликовано: 9 фев 2025
- Intro to Eigenvalues/Eigenvectors: • Using determinants to ...
Intro to Diagonalization: • How the Diagonalizatio...
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Why Diagonal Matrices are Awesome: • Diagonal Matrices are ...
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In this video we will see an example of compute eigenvalues, eigenvectors and ultimately diagonalizing a matrix when the eigenvalues are complex or imaginary numbers. We begin by setting up the eigenvalue/eigenvector formula. For a 2x2 matrix, this becomes a quadratic equation and indeed this can have complex solutions. Row reductions, gaussian elimination, row echelon form all work the exact same way to compute the eigenvectors but now we can multiply by complex numbers as well.
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Thanks so much for this video Professor! My instructor just skipped over complex eigenvalues and told us to learn ourselves, so I stumbled across this video and it’s extremely helpful. You explain things well great teaching 😄
your videos are actually incredibly well made and explained :) far better than my prof
Good stuff man, you pretty much explained it better than the other teachers i've come across.
thank you this video changed my life
Gave me a serious hand by working with differencial equantions thanks man from the internet
I watch one or two of these every month or so when I need to revisit something that's gone fuzzy
Trefor You are the man, THE Man! loving the content. You earned my subscription!
Good stuff!... working my way through the series on eigenvalues/eigenvalues/diagonalization.
Just In Time Literally!!! Thanx!!
Thanks. While this isn't hard, it is something that is often neglected. Some popular linear algebra books don't even cover this any more. I think I learned a bout this a little in ODE. In one of your videos you asked for suggestions for content many videos that plug little gaps like this. Unfortunately right now, I can't come up with other examples off of the top of my head.
So well explained. Thank you.
Thank you for doing what you do
Te agradezco que pongas subtitulos en español a tus vídeos. Saludos
Amazing video.
at 8:27, to gett the eigenvector, can't you just do -iv1+v2=0, get v2=iv1, giving us i for v2 and 1 for v1?
This was a big help! Thank you!
Nice explanation
Thanks, this helped a bunch!
dude thank you so much. After the pandemic, all my classes are now online and my professor already didn't do examples in class. This is super helpful in its own right, but even moreso considering the situation I'm in.
As a question at 7:32 if you had a pivot position in each row, is that matrix still diagonalizable (for any example)?
great stuff, just confused as to how i * -i is 1?
i*i = -1 and so i*(-i)=1
where does this prof work im gonna enroll rn
I love ur vids but amy way you can get a lavalier mic? The small collar hook mics?
Is it just a coincidence that the matrix corresponding to a π/2 rotation i as an eigenvalue, where i corresponds to a π/2 rotation in the complex plain?
Actually, it's not a coincidence!
In general, if you have this matrix that rotates every 2D vector by an angle of θ:
(cos(θ) -sin(θ))
(sin(θ) cos(θ))
then you can prove that its eigenvalues are:
λ_1 = cos(θ) + isin(θ)
λ_2 = cos(θ) - isin(θ)
which can be rewritten (using Euler's formula, e^(ix) = cos(x) + isin(x)) as:
λ_1 = e^(iθ)
λ_2 = e^(-iθ)
which correspond to rotations in the complex plane by angles of θ and -θ, respectively!
In the specific case where θ = π/2, this matrix reduces to
(0 -1)
(1 0)
and its eigenvalues reduce to
λ_1 = e^(iπ/2) = i
λ_2 = e^(-iπ/2) = -i
Perfect
Yeah so i did this but the textbook is asking that i do it in a form without complex values somehow... "Find the formulas for M^n where M is the matrix, and n is a positive int value: [[5,-3][3,5]] (Your formulas should not contain complex numbers)"
I got:
5 + 3i with corresponding vector
5 - 3i
I know it should be U*D^n*U^-1 since all the other Us and U^-1s cancel. but how can i write this in real nums?
you should have denotes the lambdas and the eigenvectors with a subscript.
I'm having some trouble visualizing this, if a vector is in 2 dimensions, are complex vectors in 4d space?
If it has two components then yes it’s in four dimensional real space aka two dimensional complex space
I'm confused on the magnitude of the eigenvector being zero.
Rad(i^2 + 1^2) = 0 right?
I have a doubt...so matrices with complex eigen values are diagonalisable ??
Great video! However, 6:59 'one plus one' is NOT zero! It is two! :-p
my goodness!
If you make this video in hindi then you will huge Views in india
How does the order of eigen vectors matter if any in P ? What happens if i swap the order does it change anything?
The order of eigen values and eigen vectors have to match. You can write it in any order you want as long as you write the corresponding eigenvalues correctly.
Can someone please explain why i * -i is 1? Or vice versa i * i is -1?
Thank you bro Id also like to know
i² = -1 , then -1* i²=1
Symmetric matrices (real eigenvalues) are dual to anti-symmetric matrices (complex eigenvalues).
Real is dual to complex -- complex numbers.
Bosons (symmetric wave functions) are dual to Fermions (anti-symmetric wave functions) -- wave/particle or quantum duality.
Bosons are dual to Fermions -- atomic duality.
"Always two there are" -- Yoda.
It'd be better if a more complicated example was used.
This doesn't work with 3*3 matrixs
You made a mistake when calculating eigenvectors, you multiplied top row by i and just left it there instead of just adding it to the bottom one. So the correct matrix is -i 1 0 0
I dont see any mistake with his work
that would be a mistake if he was using elimination to find determinants, for solving equations you can always multiply a row by a scalar
So much confusion staff