Stability and Eigenvalues: What does it mean to be a "stable" eigenvalue? 

I think it would be good to discuss the different kinds of stability, such as Lyapunov stable, asymptotically stable, exponentially stability or BIBO stable.

You and Brian gave me the motivation to study the control system. Thanks, both of you.

Finally you are the real only one who explained the eigen values in clear terms. Thank you

I watched all the videos so far! That was great and easily understandable! But I really appreciate it if you explain the concept of manifolds and their application in dynamical system solutions analysis!

Maybe also worth mentioning that a "stable" matrix, so all its eigenvalues have a negative real part, is also called a Hurwitz matrix. (The discrete time equivalent to this is a Schur matrix, whose eigenvalues lie inside the unit circle of the complex plane).

So, stability in the system as far as vibration. If you have a positive eigenvalue the vibration will increase and if negative it will decrease. When I looked at the title I thought it was talking about the eigenvalue itself being unstable ( sorta like the classical Gram-Schmidt algorithm losing its orthogonality ) . Super brain fart-- continue on! Good lesson.

This is the best video I have ever seen about this subject. Thank you so much!!

Hi Steve, can you play a bit with repeated eigenvalues? This is the last video I saw, so I don't know if in the next classes you will give such an example.

Thank you so much, Steve

This issue of the unstability is the main reason why I don't deeply believe in the concept of modes and eigenvalues as a religion. David Ruelle, in the 70's, had already seen that the concept of many many modes should be replaced by a few strange attractors. In physics, the evanescent waves exist ( lambda negatif), but nobody has never seen a wave increasing with the distance. Certainly, the concept of mode/eigenvalue should be replaced by the one of strange attractor. But nobody has never succeeded to do this. The theory of chaos has never given the slightest little engineering tool for prediction...

Nevermind, I have just seen there is a video about it in the next lectures.

I've been enjoying the series... and I feel the need to ask: How are you behind the board and writing so effectively right-to-left on your side of the board? I have been thinking about a number of film and After Effects techniques flip video footage, however in all of those that I can think of you would still have to be writing backwards. I actually think this is filmed as simply as it appears, a large sheet of plexiglass or simply glass,, a blackout curtain, and primarily lighting from the sides. The text is in the periphery of the light cone or good diffusion scrims are being used to make the light less harsh (which we can isn't entirely enough to compensate for the strong side lighting behind the plexiglass as there are strong shadows around your nose, your philtrum, and sides of your mouth while your right and left side are pretty evenly well lit, only "valleys" symmetric to both sides end up strongly shadowed). The lighting is "highlighting" the text as it's making the text appear to fluoresce slightly with the light passing through the marker ink contrasted against the black background.

Great lecture sir. Please work on some LCS lecture series also

Thank you Professor! Just for completeness: not all square matrices are diagnosable, and there is repeated eigenvalues case. Maybe it was worth to add a linear algebra primer as a side note to the lectures.

Superb! amazing lecture!

Thank you, Dr. Brunton.

I have a feeling these lectures would be helpful to understand renormalization group flow in my physics work

You will discuss Ljapunov second method in this lectures? Nothing on repeated eigenvalues?

Thank you professor...this is awesome...can you explain on how to draw Lypunov exponent with different equations using MATLAB?

Great video...really need to appreciate the efforts you are putting in💫

great teacher

What if A is dependent on time A(t)? Can I still analyze the system stability at various instants of time by analysis of the constant A(t=tfixed)? And would this stability information indicate a tendency to diverge at the time tfixed?

Thank you very much!

Sir I have problem of x2dot=sinx1-x2 + u , how check stability by lyapunov, please reply

Please explain about Lyapunov equation or test

very good

Very nice

Why at 4:01 you wrote down x(0)? Where does It come from?

yup! nice! so governed by (exp)... good luck with it! giddy up pony!

Why should Polish people sit only on the left side of an airplane? Because right-half-plane poles cause instability.

Flight dynamics and control system engineer from india

At 9:17 I hear about oscilation at frquency 'b' can someone explain what it means with a real life application example ?