It can be noted, in the context of control theory, that the difference between an A with repeated eigenvalues with A diagonalizable or one Jordan block associated with that eigenvalue can have a big influence on the controllability/observability of the system.
professor Steve, your way of teaching is the way i dream to do one day when I finish my PhD..Brilliant. I will apply for a PhD in your university just to meet you. role model..
At 29:00, I wish you would solve examples of the three types of rank to give us some intuition for how this is done and for what each implies about the Jordan canonical form.
I'm curious about one thing in JCF: if a matrix A is non-normal and has repeated eigenvalues, does it necessarily have a non-trivial JCF, i.e. is its JCF non-diagonal? Or if not,, does this property hold generically, i.e. a.e. (with respect to some reasonable measure)? The reason I am asking is that this allows for secular growth.
Can someone explain how can he make this camera view, the board is glass I assume but if he inverts the view how on earth is it possible to see him writing from left side to right? Thanks
If you were to view him from the front in real life, from the camera's perspective, you would see him writing from right to left and everything is mirrored. This is because you are seeing him from the front, his left is your right. The video is then mirrored in an editing software, which then reverses the writing to the correct direction, and of course the direction of the writing changes to left to right.
I appreciate your attempt to explain the Jordan canonical form but that's not the Jordan canonical form. Case 2 is not in Jordan canonical form. Also, Jordan form is far more intricate than you make it seem. For example, take a 3 by 3 matrix with a 1 in the (1,2) entry and a 1 in the (2,3) entry and 0's everywhere else. This is a Jordan block, but it doesn't fit any of your three cases. if you have a 2 by 2 matrix with two distinct complex eigenvalues, then the Jordan block would just be the diagonal matrix whose diagonal entries are those two eigenvalues. Unfortunately, there are a number of other issues with the content of the video. If anyone reading this wants to know what the Jordan form actually is, you can consult an advanced linear algebra book like A Second Course in Linear Algebra by Garcia and Horn.
The complex conjugate pair of eigenvalues in case 2 can be diagonalised too, if we allow a change of basis using complex matrices, but the form presented here is the best we can do if we stick to real matrices. The matrix you presented here (0,1,0; 0,0,1; 0,0,0) is a 3x3 Jordan block with eigenvalues 0, as in case 3.
It can be noted, in the context of control theory, that the difference between an A with repeated eigenvalues with A diagonalizable or one Jordan block associated with that eigenvalue can have a big influence on the controllability/observability of the system.
professor Steve, your way of teaching is the way i dream to do one day when I finish my PhD..Brilliant. I will apply for a PhD in your university just to meet you. role model..
Thank you so much for putting all these videos up for everyone to view
Another moment of stumbling across excellent content by Prof. Brunton for some random topics I found difficult to understand
At 29:00, I wish you would solve examples of the three types of rank to give us some intuition for how this is done and for what each implies about the Jordan canonical form.
Great explanation on Jordan's!
Within first 10 seconds like and comment, keep going you are the best , love your videos .
From Egypt with love
I'm curious about one thing in JCF: if a matrix A is non-normal and has repeated eigenvalues, does it necessarily have a non-trivial JCF, i.e. is its JCF non-diagonal? Or if not,, does this property hold generically, i.e. a.e. (with respect to some reasonable measure)? The reason I am asking is that this allows for secular growth.
I'm a little confused by the first example- why is it heading to null space? isn't it the identify matrix?
Can someone explain how can he make this camera view, the board is glass I assume but if he inverts the view how on earth is it possible to see him writing from left side to right? Thanks
he's left-handed
If you were to view him from the front in real life, from the camera's perspective, you would see him writing from right to left and everything is mirrored. This is because you are seeing him from the front, his left is your right.
The video is then mirrored in an editing software, which then reverses the writing to the correct direction, and of course the direction of the writing changes to left to right.
Thank you.
Great video
Thanks!
Edit: I paused at 20 minutes, wrote a question, un-paused, and immediately had it answered, whoops 😓
I appreciate your attempt to explain the Jordan canonical form but that's not the Jordan canonical form. Case 2 is not in Jordan canonical form. Also, Jordan form is far more intricate than you make it seem. For example, take a 3 by 3 matrix with a 1 in the (1,2) entry and a 1 in the (2,3) entry and 0's everywhere else. This is a Jordan block, but it doesn't fit any of your three cases.
if you have a 2 by 2 matrix with two distinct complex eigenvalues, then the Jordan block would just be the diagonal matrix whose diagonal entries are those two eigenvalues. Unfortunately, there are a number of other issues with the content of the video.
If anyone reading this wants to know what the Jordan form actually is, you can consult an advanced linear algebra book like A Second Course in Linear Algebra by Garcia and Horn.
The complex conjugate pair of eigenvalues in case 2 can be diagonalised too, if we allow a change of basis using complex matrices, but the form presented here is the best we can do if we stick to real matrices.
The matrix you presented here (0,1,0; 0,0,1; 0,0,0) is a 3x3 Jordan block with eigenvalues 0, as in case 3.