Another great video. I know this can be tedious for you, Dr Brunton, but please remember that is what makes these videos soooo valuable. Non-geniuses like myself can actually follow along, or rewind if needed, unlike classroom lectures where we spent endless hours lost and bored. I actually learn from your awesome lectures. Please stay patient, don’t skip steps, and keep stating the obvious…because if it was obvious, there would be no reason for this type of education. Thank you so much for this!
For future ref for beginners) 1. (28:15 and 35:40) The arrows in the figures should be clockwise direction. 2. v=2y (not v=y) if the matrix is [[0,2], [-2,0]] 3. (32:12) actually we don't need to calculate the 2nd order eq., but we can just add -1 to the previous lambdas(+/-2i). (because A-I*lambda=0)
Amazing way of explanation Prof. Brunton. Keep posting more and more such interesting, insightful, and pedagogical lectures for the present and future generations
This lecture was awesome... thank you.❤❤❤ At 32:13, the determinant of matrix [-1-lambda 2; -2 -1-lambda] is equal to (-1-lambda)^2 + 4 = lambda^2 + 2lambda + 1 + 4, which equals lambda^2 + 2lambda + 5...
In Steve’s example at the end of the video, the characteristic equation is: lamda squared + 2*lamda plus 5 = 0, giving the complex solutions: -1 plus or minus 2i, as expected.😊
"this might be the point where you speed me up to 1.5 times and I go chipmunk... I've had a lot of coffee so that might happen already...". This is great math and comedy (that quote starting at 10:23).
At the 28:00 minute mark, The phase diagram has the arrows pointing counter clockwise. Shouldn’t they be going clockwise? I.e, when the mass is pulled to a positive X, shouldn’t the velocity start in the negative Y direction, not positive Y.
Thank you so much Steve, this is the best tutorial about ODE I have ever seen, very detailed and instructive. One quick question: at 26:32, double derivative of x equaling -4x actually renders A = {{0, 1}, {-4, 0}}, not the matrix you gave at the beginning A = {{0, 2}, {-2, 0}}. Even though they have the same eigenvalues (±2i), their eigenvectors are different, and the final expressions for x(t) and y(t) are different. I feel a bit puzzled, do you mind giving me some hints?
I noticed that the originial equation in vector and matrix form has 1st order derivative in time, while the (rotating) solution looks like a solution to 2nd order ODE (and not exponentially blowing up/decaying as 1st order ODE solutions often do). fascinating. wonder what happens when we do have a matrix differential equation 2nd order in time
Dr. Brunton, can we tell the overall shape of the phase portrait from the shape of the matrix A in the equation xdot = Ax (where A is 2x2 matrix and x a column vector) . Does whether or not A is diagonal/anti-diagonal, whether or not the non-diagonal entries are real/imaginary/zero matter?
There is the hairy ball theorem that guarantees that there's always at least one point on Earth where the wind isn't blowing. I'm wondering if we could also explain it with an imaginary eigenvalues approach ...
is it possible that the phase portrait arrows are inverted, should go clockwise not counterclockwise ? if y is the xdot state then positive y should go towards positive x. i am thinking that happened due to the blackscreen inversion maybe?
Thank you a lot for your videos. I just think that you are making a mistake in the direction in which the system is rotating. I think it is supposed to rotate clockwise instead.
Great explanation about physical intuiation of dynamic systems! But for the model [-1,2;-2,-1], how does it map to a spring oscillator when dx/dt=-x+2v from the first line, but dv/dt=-2x-v looks physical to me.
to all the learners: when he effed up his own example at the end? THAT'S WHY YOU GO STEP BY STEP AND DON'T DO 20 STEPS AT ONCE :P. even when you're good at this stuff, combining steps leads to tears - go step by step folks!
look out! I just found my bellybutton! super duper! love his teachings!! good luck! I'm just waiting 2 see Steve, break out the plywood and make furniture! this might blend well in *rhino-grasshopper* . something like matlab..good luck all !! he makes superb use of mathematical principals! I had always used *imaginary* as fractions. where in a nvidia v100, will see this as FP16! symmetric!🥳
![Blox Fruits Dragon Rework Update [Full Stream]](http://i.ytimg.com/vi/EqDAp8udhm0/mqdefault.jpg)
Another great video. I know this can be tedious for you, Dr Brunton, but please remember that is what makes these videos soooo valuable. Non-geniuses like myself can actually follow along, or rewind if needed, unlike classroom lectures where we spent endless hours lost and bored. I actually learn from your awesome lectures. Please stay patient, don’t skip steps, and keep stating the obvious…because if it was obvious, there would be no reason for this type of education. Thank you so much for this!
This is really great video. Thank you very much.
BTW: 32:20 It's L^2+2.L+5.
For future ref for beginners)
1. (28:15 and 35:40) The arrows in the figures should be clockwise direction.
2. v=2y (not v=y) if the matrix is [[0,2], [-2,0]]
3. (32:12) actually we don't need to calculate the 2nd order eq., but we can just add -1 to the previous lambdas(+/-2i). (because A-I*lambda=0)
That's exactly true. With the (second-order) differential equation x''=-4x, we get by calling y=x'/2 the linear system
(x' y')=[[0 2], [-2,0]](x y)
Amazing way of explanation Prof. Brunton. Keep posting more and more such interesting, insightful, and pedagogical lectures for the present and future generations
This is art. Beautiful
I cannot BELIEVE how underrated you are. You deserve to have millions of subscribers
This lecture was awesome... thank you.❤❤❤
At 32:13, the determinant of matrix [-1-lambda 2; -2 -1-lambda] is equal to (-1-lambda)^2 + 4 = lambda^2 + 2lambda + 1 + 4, which equals lambda^2 + 2lambda + 5...
In Steve’s example at the end of the video, the characteristic equation is: lamda squared + 2*lamda plus 5 = 0, giving the complex solutions: -1 plus or minus 2i, as expected.😊
Sorry - my comment relates to Steve’s previous video on centres, not this video on Stability and Eigenvalues.
"this might be the point where you speed me up to 1.5 times and I go chipmunk... I've had a lot of coffee so that might happen already...". This is great math and comedy (that quote starting at 10:23).
This is beautifully quantum gate in Quantum information science and a squeezing operator in Quantum photonic. Beautiful!
I am referring to the rotation matrix. And as beautiful as this is, the position and velocity are conjugate variables in QIS
At the 28:00 minute mark, The phase diagram has the arrows pointing counter clockwise. Shouldn’t they be going clockwise? I.e, when the mass is pulled to a positive X, shouldn’t the velocity start in the negative Y direction, not positive Y.
Correct. The velocity will be initially negative so with arrows clockwise
Thank you so much Steve, this is the best tutorial about ODE I have ever seen, very detailed and instructive. One quick question: at 26:32, double derivative of x equaling -4x actually renders A = {{0, 1}, {-4, 0}}, not the matrix you gave at the beginning A = {{0, 2}, {-2, 0}}. Even though they have the same eigenvalues (±2i), their eigenvectors are different, and the final expressions for x(t) and y(t) are different. I feel a bit puzzled, do you mind giving me some hints?
Let v = 1/2 * \dot{x}. Then \dot{x} = 2v, and \dot{v} = 1/2 \doubledot{x} = -2x.
Thank you so much, professor. Waiting for the next video
Thanks Mr. Bruton!
I noticed that the originial equation in vector and matrix form has 1st order derivative in time, while the (rotating) solution looks like a solution to 2nd order ODE (and not exponentially blowing up/decaying as 1st order ODE solutions often do). fascinating.
wonder what happens when we do have a matrix differential equation 2nd order in time
great lecture!!!!!!
If the eigenvectors are normalized to lenght 1, the inverse of T is simply the transpose T’, making inversion process easier
Dr. Brunton, can we tell the overall shape of the phase portrait from the shape of the matrix A in the equation xdot = Ax (where A is 2x2 matrix and x a column vector) . Does whether or not A is diagonal/anti-diagonal, whether or not the non-diagonal entries are real/imaginary/zero matter?
Amazing.
There is the hairy ball theorem that guarantees that there's always at least one point on Earth where the wind isn't blowing. I'm wondering if we could also explain it with an imaginary eigenvalues approach ...
is it possible that the phase portrait arrows are inverted, should go clockwise not counterclockwise ? if y is the xdot state then positive y should go towards positive x. i am thinking that happened due to the blackscreen inversion maybe?
at 5.08 can I re write the equation as d/dt [ v x] = [-1 0; 0 1] [v x], so that the A matrix become a diagonal matrix?
The A matrix is actually [0 -1; 1 0] which is not a diagonal matrix either
Thank you a lot for your videos. I just think that you are making a mistake in the direction in which the system is rotating. I think it is supposed to rotate clockwise instead.
Great explanation about physical intuiation of dynamic systems! But for the model [-1,2;-2,-1], how does it map to a spring oscillator when dx/dt=-x+2v from the first line, but dv/dt=-2x-v looks physical to me.
same question....did you figure it out?
to all the learners: when he effed up his own example at the end? THAT'S WHY YOU GO STEP BY STEP AND DON'T DO 20 STEPS AT ONCE :P. even when you're good at this stuff, combining steps leads to tears - go step by step folks!
huh so the imaginary part of the eigenvalues turns out to be the angular frequency
oh its in phase space!
Thank you for sometimes doing the calculations wrong. It forced me to actually think through the math instead of idling accepting it.
look out! I just found my bellybutton! super duper! love his teachings!! good luck! I'm just waiting 2 see Steve, break out the
plywood and make furniture! this might blend well in *rhino-grasshopper* . something like matlab..good luck all !!
he makes superb use of mathematical principals! I had always used *imaginary* as fractions. where in a nvidia v100, will see this as FP16!
symmetric!🥳