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 😄
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.
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
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?
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)?
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.
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
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.
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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.
Good stuff!... working my way through the series on eigenvalues/eigenvalues/diagonalization.
thank you this video changed my life
I watch one or two of these every month or so when I need to revisit something that's gone fuzzy
Gave me a serious hand by working with differencial equantions thanks man from the internet
Trefor You are the man, THE Man! loving the content. You earned my subscription!
great stuff, just confused as to how i * -i is 1?
i*i = -1 and so i*(-i)=1
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
Amazing video.
where does this prof work im gonna enroll rn
Te agradezco que pongas subtitulos en español a tus vídeos. Saludos
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?
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
Nice explanation
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 have a doubt...so matrices with complex eigen values are diagonalisable ??
I'm confused on the magnitude of the eigenvector being zero.
Rad(i^2 + 1^2) = 0 right?
Thanks, this helped a bunch!
This was a big help! Thank you!
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?
I love ur vids but amy way you can get a lavalier mic? The small collar hook mics?
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)?
you should have denotes the lambdas and the eigenvectors with a subscript.
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
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.
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
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
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.
So much confusion staff