Love your content! I've been watching many videos on differential equations as I'm cramming for a final and all of yours has easily been the most helpful
im confused on how you went from det(a-L I) to (L+1/2)^2 +1 (L: lambda). wouldnt it be ((-1/2)-L)^2? if you use -A, then you get this but you also get -1? I'm confused about this step.
can you please clarify the last part....for the solution of x2 wont we solve the equation again but with the conjugate of the complex eigen value we obtained?
@@BallinLongboarder209 I'm wondering if the imaginary part of the first eigenvalue is equal to the real part of the second eigenvalue, but I'd like to hear the professor's explanation.
Note that both two eigenvalues have -1/2 in their real parts. e^(-t/2) pops up when you plug in these eigenvalues into original assupmtion; x(t)=Ve^(lambda t)
Equal eigenvalues could mean that the matrix is a scalar multiple of the identity matrix, and all vectors with real components are eigenvectors. The phase portrait of such a system is a star pattern, where all solutions follow straight line paths, with a radial position that is an exponential function of time. Equal eigenvalues could also mean that there is only one eigenvector, and the phase portrait is called an improper node. Every path toward the origin, for equal negative eigenvalues, will approach the equilibrium position on a path that asymptotically approaches a direction parallel to the only eigenvector.
@@ifraawan7538 Complex eigenvalues means that there is no real vector that remains in the same direction, after being operated upon by the matrix. All vectors will be rotated (and scaled if applicable), if there are complex eigenvalues. Real eigenvalues means there are special vector directions that remain constant after the matrix operates on them, and that only a scaling takes place. That vector is the eigenvector, and the amount of scaling is the eigenvalue, which is what would happen for real eigenvalues and real-components of eigenvectors. If used for a solution of a diffEQ system, complex eigenvalues mean that the solution is a linear combination of sine and cosine, each multiplied by an exponential envelope. The real part of the complex eigenvalue is the exponential growth/decay rate, and the imaginary part is the frequency of the trigonometric functions. A negative real part of a complex eigenvalue, means a decaying spiral toward the equilibrium point. A positive real part of a complex eigenvalue, means a growing spiral away from the equilibrium point. A purely imaginary eigenvalue means an ellipse (or circle in a special case) that circulates the equilibrium point in a stable path that remains the same.
Find other Differential Equations videos in my playlist ruclips.net/p/PLkZjai-2JcxlvaV9EUgtHj1KV7THMPw1w
Thank you, Mike Ehrmantraut
ahahahahahaha i thought it was mike too
fingerrrrrrrrrrrrrrrr
I thought he was professor Fletcher
Love your content! I've been watching many videos on differential equations as I'm cramming for a final and all of yours has easily been the most helpful
Thank you so much for your videos, all of them were really helpful! Now I regret not having chosen Hong Kong for my engineering courses.
Excellent, very impressive
Thank you so much. Until now it was too confusing for me..U made it clear..
thank you prof. you precise explanation of concepts without reservations.
OMG this is very good. Thanks a lot
im confused on how you went from det(a-L I) to (L+1/2)^2 +1 (L: lambda). wouldnt it be ((-1/2)-L)^2? if you use -A, then you get this but you also get -1? I'm confused about this step.
hello, Thank you so much for your video. It helps me a lot. It is really helpful.
Thank you very much for all your amazing videos. You explain concepts clearly in ~10 minutes that would take up the entire lecture period in class.
can you please clarify the last part....for the solution of x2 wont we solve the equation again but with the conjugate of the complex eigen value we obtained?
im confused about this as well, isnt this only half of the solution?
@@BallinLongboarder209 I'm wondering if the imaginary part of the first eigenvalue is equal to the real part of the second eigenvalue, but I'd like to hear the professor's explanation.
@@MrScreaney I might be able to give you an answer tomorrow. I think it's enough to construct a general solution with one eigenvalue.
When eigenvalues are complex conjugates of each other, you can choose either the first one, or the second one, won't be wrong in any case
Thanks so much sir! Very clear and straightforward!
Super helpful! Thank you so much!!
THANK YOU SO MUCH! Your's are the only videos I've found which explain this and work through an example in a way I can understand!
thank you sir
Can someone explain to me how he dragged out the e^(-t/2) for both the real and imaginary part???
Note that both two eigenvalues have -1/2 in their real parts. e^(-t/2) pops up when you plug in these eigenvalues into original assupmtion; x(t)=Ve^(lambda t)
should we find c1 and c2 is it possible
ermantraut without the stache. thats goals
Plz upload for equal eigen values also🙏🙏
Equal eigenvalues could mean that the matrix is a scalar multiple of the identity matrix, and all vectors with real components are eigenvectors. The phase portrait of such a system is a star pattern, where all solutions follow straight line paths, with a radial position that is an exponential function of time.
Equal eigenvalues could also mean that there is only one eigenvector, and the phase portrait is called an improper node. Every path toward the origin, for equal negative eigenvalues, will approach the equilibrium position on a path that asymptotically approaches a direction parallel to the only eigenvector.
Hello Chasnov, i had a question. What would be the outcome whem we will calcute for the nagative value of of Lambda such as -1/2-i?
Spiral in to the origin
I need only complex eigenvalues
we want to know what is complex eigenvalues introduction and properties etc so please after that u give example
so please reply my answer
@@ifraawan7538 Complex eigenvalues means that there is no real vector that remains in the same direction, after being operated upon by the matrix. All vectors will be rotated (and scaled if applicable), if there are complex eigenvalues. Real eigenvalues means there are special vector directions that remain constant after the matrix operates on them, and that only a scaling takes place. That vector is the eigenvector, and the amount of scaling is the eigenvalue, which is what would happen for real eigenvalues and real-components of eigenvectors.
If used for a solution of a diffEQ system, complex eigenvalues mean that the solution is a linear combination of sine and cosine, each multiplied by an exponential envelope. The real part of the complex eigenvalue is the exponential growth/decay rate, and the imaginary part is the frequency of the trigonometric functions.
A negative real part of a complex eigenvalue, means a decaying spiral toward the equilibrium point.
A positive real part of a complex eigenvalue, means a growing spiral away from the equilibrium point.
A purely imaginary eigenvalue means an ellipse (or circle in a special case) that circulates the equilibrium point in a stable path that remains the same.
repeated roots?