The Jacobian Matrix

hello everyone and welcome to another video so today I want to spend a couple of minutes talking about the Jacobian matrix and I know that name might sound a little bit intimidating but really the idea behind it is surprisingly simple we're going to see that the Jacobian matrix is really nothing more than taking the idea of a derivative extending that to a gradient and then extending that again to what's known as this Jacobian matrix and this Matrix actually shows up a lot in science and engineering because we're going to see at the end of the day what this thing does is it allows us a way to calculate and quantify the Sens it of a particular function element to perturbations in a given independent variable so that might sound like a mouthful but if we take uh a crawl walk run approach to this I think you'll see that uh this is actually pretty pretty consumable so to start with our crawl phase let's jump back to the idea of a derivative right derivative is surprisingly simple right we all remember this from high school calculus if we have some function f like this and it takes in some input X and does something to it to produce an output in this case what it does is subtracts two from that value it raises it to the power three and then multiplies by three right so again you can think about this from a functional perspective as a function which takes in a single independent variable X and it spits out F ofx right okay so you might ask yourself okay this is what this function is we can easily plot this this is nothing uh Earth shattering right I'll plot F ofx versus X right and it looks something like this okay and now the next logical step might be asked what is the derivative of this function f and again don't overthink this right you can break out your old high school calculus textbooks right and you can say okay the derivative of this function it's just this right um now the only thing that maybe we should mention is we're going to write this like this right the function f it's only a function of a single variable so I don't need partial symbols I can just use a d right because it's only a function of one variable so d FDX is just this and what I want to make a note of is notice here right that this is still a function of the independent variable or the input X right so to make this explicit what I want to write and a lot of times you'll see this is something like this right I'm going to add this notation just to remind ourselves that the derivative of the function it's still potentially a function of the independent variable and in fact when you plot this I think you'll see that yeah that's definitely the case it's just some other function of X right as X changes the derivative value changes right and again nothing Earth shattering because what this is doing right is everyone knows the derivative is basically telling you the slope of the original function f right so if I look at different points along this or a different input values of X like let's say I choose some x value out here like at uh I don't know minus 0.75 right so what this derivative is telling me and sorry I didn't draw this very well I should have probably extended this a little bit more right is what this is telling me is I can read this value here and that's telling me effectively the slope or how fast is this function f changing at this particular x value so if I'm sitting here if the input to this function is 0.75 right and now I perturb it by some little amount like let's add on a 0.1 right what this derivative is telling me is how much is this perturbation basically Amplified in the output right because this little 0.1 perturbation here it's going to result in some type of perturbation in the output right so it's basically a specific run is going to produce a rise right so you can see this that obviously this sensitivity of this function changes as you change the value of x right so for example if I choose another value x = 2 right over here here right we see that it actually doesn't change here right the slope of this function is completely flat here meaning if you're sitting here at two if the input to this function is now two right and you put in a little bit of a 0.1 or the tiniest little input perturbation you don't expect the output to change at all right that's what this is telling you and again you can change this so let's see red um oh sorry I probably should have made this green sorry to be consistent with uh sorry I I got too excited with the colors tell you what let's leave it like uh no let's change this let's make this green cuz later on I I have another example okay so this should have been green two because I want to go red green blue just to kind of be consistent okay there we go okay so there's my green value and then let's choose a blue value over here you know something uh what did I pick here 3.25 right here's some other blue value and again you can read this up here to get the slope of the function or the sensitivity of the function there right so you can now ask yourself okay if the function has an input of 3.25 and I put some little tiny perturbation on it how much does the output change right so again all the derivative is giving you right the derivative one physical way to think about this is sure it's the slope of the function but really physically what that's telling you is how much does the function change when you change the input or you perturb it a little bit away from a given input value right that's what this slope is effectively giving you it's right it's how sensitive is the function to changes at this particular particular location right so that's what the derivative tells you and obviously as we discussed it changes depending on where X is right okay so that's the idea of a derivative um let's extend this idea now to the idea of a gradient okay so remember the idea with the gradient now is let's still go ahead and consider a scalar function meaning the function still outputs a single variable but now you could have potentially multiple inputs or multiple independent parameters or variables feeding into the function right so in this case let's say there are n different inputs or independent variables to this function X1 X2 all the way down to xn right now if you remember our previous video where we discussed the gradient we know that the gradient then of this given function f which has these inputs it's just given as this right it's written sometime as DF or nobla F the other alternative notation for this is it's now it's a partial of this function f with respect to this Vector X okay and what that actually means right is it means again you got to remember that this Vector this xar is a vector right there are n independent inputs right so what this notation means is it just means the partial of f with respect to X1 stack that on top with partial of f with respect to X2 all the way down stacked at the bottom with partial of this function f with respect to xn right and again we got to remind ourselves and remember that I and others like the notation that you got to remember that by the time you take these partials this is likely still a function of the independent variables X so again I'm going to put the notation like this right so the gradient of f is a function of X it depends on where you are just like we saw here right its value depends on what the value of x is and again I'm going to put this notation up here as well right and actually what we should probably do is we should put this notation here as well right because every single one of these components is going to be a function of the input variable or the input Vector X all right so at the end of the day you end up with an N by1 Vector which represents the gradient okay and again depending on which uh reference you're looking at or what author some people like to stack it the gradient Vector uh uh sorry stack the vector as a column Vector like like I'm drawing here this is what I prefer some other people like it as a row Vector um but again just be careful of what notation is being used and um be consistent right so I like to stack this up vertically a lot of people do so you'll see the gradient written as a vertical column Vector okay so that's the idea right so again let's look at a concrete example just to drive this home in fact in this gradient video right where we discussed this gradient in a little bit more detail this is the example function that I used so just to refresh your memory right the way we can visualize this again it's still a scalar function right in the sense that it still outputs a single number right the way it computes the single number though is this function has two inputs now it has an X1 and it has an X2 right two independent variables or independent parameters however you want to think about this these two inputs produce a single output right which is f of x1x2 right and here's the algorithm or the formula of how it does this right um so to compute the gradient of this particular function right I'm just going to use my expression or my equation over here all I got to do is the first entry is the partial of this function with respect to X1 while holding X2 constant or treating X2 as a constant the second element is the is the exact opposite right I'm going to take the partial of this function with respect to X2 while holding X1 constant so what you end up with here is I think everyone can see you can take the partial derivatives of these two you'll get these two expressions right so again what this in physically means is let's go ahead and um actually I'm going to spin a movie over of this function on the side um what you're seeing in this picture is the orange surface is obviously this function for a whole bunch of different X1 and X2 values and as you can see right the slopes uh of this function change depending on where you are so again maybe what we'll do is let me use our same red green blue idea so depending let's choose a red Vector X of in this case I think what you're seeing here is these red dots let me just make sure I've got this correct right I used uh Min -3 for X1 and positive2 for X2 and then the Green Dot you're seeing here is another x value of I think this is actually 0 0 okay and then the blue one that I'm drawing on that surface here is 12 0 okay so at these different locations right these numbers change right so the gradient changes right and what you can see from this picture is what the gradient is actually telling us right physically what comes up right is this first element it's telling you the sensitivity of this function f to changes or perturbations in only X1 so if you move in the X1 direction right how much does that do you expect that to change the output of the function right does the function grow quickly or slowly does it go positive or negative right that's what this first entry of the gradient is telling you right it's the sensitivity of the function to pertubations in the first independent variable right the second element of the gradient right is is very similar right it's the sensitivity the function to changes SL perturbations in X2 considering that you hold X1 constant right so again depending on where you are if you're at that red green or blue dot in that pcture in in the movie you're seeing on the side here right you can see that the function changes um at different rates at those different locations and depending if you're moving in the X1 or the X2 direction right so that's what the gradient is is measuring right so really if you come back to this idea of the derivative right it's nothing more the gradient is just a multi-dimensional derivative right that's all that's telling you right both of these things right all these derivatives are telling you right either it's the total derivative here or a partial derivative over here it's just telling you the sensitivity of the function how is the output of the function going to change as you change one of these independent parameters so that's the idea with the gradient let's see let's let's leave this uh discussion of the gradient up on this side of the board let me erase this over here to get a little bit more space because all we're going to do now is extend this idea of the gradient to build the Jacobian matrix all right so now we've got the Jacobian matrix discussion and all the Jacobian matrix is is now instead of dealing with a scalar function let's talk about using a vector valued function meaning that now this function instead of having a scalar output it has a vector output so in this case I've drawn three and one way to visualize this or to think about this is that this Vector valued function it's nothing more than in this case three scalar V uh scalar functions stacked on top of one another right so that's all this thing is you can take a look at this stack these three up and now let's call this Orange Box right it's still it's a function which takes two independent variables or two inputs and now outputs three things okay so what we can do is we can just call this now let's call it F Bar of xar right so again the notation here is that this function f the bar on top if we contrast that with over here is the bar on top means that the output right this is a vector function in the sense that now there are 1 2 3 the output is a vector and the input X is still a VOR Vector just like we had over on this side we called this F of xar okay so that's what we've got it's nothing more than three scalar uh functions stacked on top of one another so if you want to think about this we can write this as defining okay F Bar of xar is our Vector valued function which is nothing more than a certain number let's call it maybe M Mike m number of these different scalar functions stacked on top of one another so you could have F1 of X F2 of X all the way down to F M of X something like this okay so that's all this function is right the vector value function is nothing more than M scalar function stacked on top of one another okay so with with this in mind now we can ask ourselves the exact same question we've been asking ourselves earlier is how does this function FB how is it sensitive to changes in different parameters or different input variables X1 X2 all the way down to xn okay so that's all the Jacobian matrix is so the jobian Matrix is just a partial set of derivatives asking how do each one of these scalar functions fub1 FS2 all the way down to FM change as a function of how these input variables uh input parameters change so if you think about this stare at this first one right here okay this is exactly what we just did correct so all that we want is basically the gradient of function F1 that already told us how function F1 is sensitive to perturbations in X1 and X2 correct so all you need is uh again let me use this other notation I'm just going to write DF bar DX bar right same same idea except now I've got instead of a scaler F I've got a vector F and all this thing is okay is we can stack up and put the gradient in the first row now remember I was stressing earlier that in in this notation we are considering the gradient to be a column Vector what we're going to do right here is we're going to knock this over on its side and actually make it a row Vector okay so again you just going to be a little careful depending on what notation or what reference you're using so in this case what we want to write here is we want the gradient of fub1 of X okay then you have the gradient of F2 of X all the way down to gradient of f f m subx and again what we have to do is we have to transpose each one of these to take the column Vector knock it over on its side make it a row Vector like this okay and again let's make sure and remind ourselves we're going to use this notation that this Jacobian matrix it's a function of where the inputs are or what inputs are going into this right so we should use consistent notation just like we did over here with the gradient right because all we see is the Jacobian matrix it's nothing more than M gradients stacked on top of one another that's all it is okay and in fact we want to be explicit and write this out let me go ahead and erase some of this see if we can fit all of this onto one board let me get get ourselves a little bit of room I think we can make this work okay so if you look at this long enough let's go ahead and just get this first row right it's nothing more than the gradient of FS1 with uh the gradient of FS1 right and we said the gradient is nothing more than the the partial der of the function with respect to each one of the independent parameters so this is partial of F1 with respect to X1 okay and again let's make a note that this is a function of where you are then you have partial of F1 with respect to X1 and it's still a function of where you are all the oh sorry X2 okay all the way down to partial of fub1 with respect to xn right because they're are n independent variables okay or n inputs to this function okay that's the first row it's just the gradient of F1 knocked over on its side so the second is similar right it's just partial of F2 with respect to X1 partial of FS2 with respect to X2 all the way down to partial of FS2 with respect to xn and you repeat this all the way down to the last row which is now partial of FM with respect to X1 then you get partial of FM of X with respect to X2 all the way down to partial of FM with respect to xn there you go okay so what we end up with is this entire thing is now an M by n Matrix okay because we see it's nothing more than M gradient vectors laid out on their side in row vector format okay so that's all the Jacobian matrix is it's basically a giant Matrix of all the mixed partial derivatives of all of the these functions and in fact maybe what we should do is we should write this down in the sense that the Jacobian matrix we see here it's an M by n Matrix okay but the row I column J okay what this tells you is it is basically it's the sensitivity right it's whoops let me write this down sensitivity of function I outut right to the change in XJ right so and in fact maybe the the better way we should have said this it's really it's the entire function it's the function f let's let's rewrite this maybe in a little bit more clear fashion right it's a sensitivity of f Bar's I output to the change in XJ okay so if I want to understand how the third output this third output responds to changes in the first input okay that would be located in this Jacobian matrix Row 3 column 1 right that that's how this works so again the Jacobian matrix it basically tells you the complete sensitivity or all the slopes if you want to think about it that way of all of these different scalar functions and how they change so to help visualize that maybe let me let me stand over here I'll try to put a picture over on the other side of the board where you can visualize each one of these in this case there are basically two dimensional functions right in sense that there are two input parameters and each one of them produce a single output and that's what I'm plotting over on the side you can see in the red green and blue they're just different surfaces so they have different sensitivities depending on how you change X1 and X2 you're you're basically walking in different directions on either the red surface for this first one the green surface for the second one or the blue surface for for the bottom one so all the Jacobian matrix is basically capturing is every single derivative and as we see you can be anywhere on that surface depending on what your X1 and X2 values are so this Jacobian matrix changes depending on the values of X1 and X2 just like how we saw the gradient Vector changes depending on what the inputs were and just like how we saw a derivative changes depending on what its input was so again we now see the whole picture right we start with the single derivative we then move and extend that idea to a gradient and then we move and extend a gradient idea now to the Jacobian matrix so this is going to be a pretty powerful tool um some history behind it the Jacobian matrix it's named after Carl Gustaf Jacob jacobe you can see over here this is what he looked like he was uh born in 1804 died in 1851 um he was a German mathematician who made some pretty fundamental contributions to things like elliptical functions um Dynamics which we're actually going to take a look at in just a second um differential equations uh in fact he there was a the the famous name that sometimes you'll hear is the Hamilton jacobe equations which is basically an alternative way to express um equations of motion for dynamic systems um and in fact a kind of fun note there's actually a crater on the near side of the moon named after him and as we see in this case there's also the Jacobian matrix which is named after him so with that history aside maybe what we should do is let's let's go get into an example and in fact why don't we use this picture we've got right here we've got these three functions this is as good as a set of functions as any let's go ahead and compute the Jacobian of this function which has two inputs and three outputs and the way those inputs and outputs are computed are given these three function so we have enough information at this point to go ahead and compute the Jacobian matrix by just taking all of these partial derivatives so let me clear off the board and let's do that next all right so let's go ahead and compute the Jacobian for this orange function so I've just written it down um as we saw it's just the the gradient of the first function laid on its side the gradient of the function of the second function laid on side and the gradient of the third function laid on its side so it's all of these mixed partial derivatives here so for example let's look at the one one element it's the partial of F1 with respect to X1 right so all you got to do is take here's F1 take this partial with respect to X1 and as you can see that gets us what just 6 X1 so that's this element here right and then the one two element it's the partial of fub1 with respect to X2 so again still keep your eye on this first function F1 take its partial with respect to X2 2 and you can see that just turns into what 3x2 2 which is that element right so you just go through this and now moving down to the second row it's now you repeat this operation now for the second function with respect to X1 and you get this right by just taking this partial and blah blah blah right so at the end of the day what we see is you end up with this 3x 2 Matrix right and again its physical interpretation of what this Matrix is is each element of this Matrix right it's measuring the sensitivity of the I output to the change in the J input so in other words if we look at the input outputs let me see if I can do this right if I make a little table if I want to look at what the the sensitivity of the first function output is to the first input variable that's the one one element right here correct and then this one two element tells me the sensitivity or the slope effectively of the first function's output to perturbations in the second input all right and etc etc you keep moving down so the the sensitivity of the second function with respect to the first input is there and then the sensitivity of the second function to the second input is here and then similarly you just kind of you can pick off each element of this Matrix right and what it's basically Al telling you right is that Matrix is fully characterizing every possible combination of the output to input sensitivity right so now you can pretty much fully characterize this orange Vector valued function we understand now if you perturb or change the inputs X1 and X2 how is that going to change each one of these three outputs right and that is exactly what the Jacobian matrix is giving us right so this is a nice abstract mathematical form formulation and example um let's look at an example of how this might show up in an actual engineering application all right so let's look at an example of an engineering system uh namely a dynamic system uh as many of you know I'm a controls engineer and a lot of times the classic example that controls Engineers love to look at for a nonlinear um set of ordinary differential equations is a classic pendulum right it's just a pendulum which is swinging about some point up here it has some Mass on the end of it and that's it it's just swinging and as we've seen in the past this admits a set of nonlinear Dynamics to make this a little bit more interesting um I've tried to spice up this pendulum a little bit so let's assume that instead of this just being like a simple boring pendulum maybe this is a test stand with a two uh bidirectional rocket engine down here where you can fire the engine in either direction and that is going to create an a propulsive force um that I'm denoting as is f engine and as this thing is swinging around due to this Engine Force there's going to be some drag on it some aerodynamic drag so I've denoted this as FD okay um as we've seen there's going to be obviously gravity pulling down on the pendulum which is going to impart some kind of moment and then the only other moment we're going to add on this test stand is let's say that there's a break up here that an operator can turn on and off so the operator can really control two things they control the throttle to the engine that will basically influence this Engine Force they can also control the break between like zero meaning no break and one meaning full break or something like that and that's going to go ahead and control how much it tries to stop this whole test stand from spinning around right and again here's some of the relevant geometry it's a length L and then as you can expect given the geometry this moment arm here is L sin Theta again this is a pretty standard problem at this point so let's just walk through it and again some of these details are just I pulled it out of thin air just to kind of illustrate R um how this might generate some set of equations of motion so again I'm going to assume that the engine thrust is you know it's maybe nonlinear depending on the throttle setting between I don't know maybe like a negative one means Back It Up In Reverse a positive one means go forward but it looks like this it's some coefficient Alpha times U1 cubed okay that's that's the engine thrust the uh aerodynamic drag let's just assume that this is something really simple it's some coefficient times the velocity of the vehicle or this engine so it's basically it's it's a linear sort of viscous uh type of damping you've got here so again velocity you could write that as just L * Theta dot okay and then the breaking moment I you know again I I made this up it's some breaking coefficient gamma times how much break is being applied by the operator between zero and one and then it's going to matter how fast it's spinning if this thing is spinning quicker The Brak is going to be more effective and if the if it's not spinning at all like Theta dot is zero the breaking moment is going to be zero no matter what the operator puts on it so again I'm just I'm picking this out of thin air it's not super Germain to the problem but I just want to have some somewhat quasi realistic um reasonable set of Dynamics okay so if you've got all these forces on moments on this thing let's just go ahead and sum up all of the moments right so you've got the moment due to the engine f * L you've got the moment the retarding moment due to drag the retarding moment due to um due to the break and then this term here is the moment due to gravity which is either going to impart a positive or A negative moment depending on S of theta right how this term looks okay so this is our moments so let's go ahead and apply Newton's Second Law right so Newton's second law for rotation right it's the moment of inertia times the angular acceleration is just the sum of the moment so I'm going to go say dot dot dot and skip a couple of steps if you're interested in all the nitty-gritty details actually for this or for any other part of this discussion right right check the link in the description of this video where you can download my um PDF set of notes where I've got uh pretty much line by line of how to get here but I'm going to assume that people are comfortable with developing equations of motion so you basically get this right this is your equation of motion so now let's choose a state and control Vector for this um in this case the state Vector is just going to be Theta and Theta Dot and then the control Vector the two controls that the that the user would have is you want U2 right so it's the engine throttle and the brake command okay so if you've got these two we can now take a step back and say okay this is how I would formulate the problem as a controls engineer but as a mathematician wanting to look at this from sort of a sterile abstract mathematical point of view both these control vector or sorry the state vector and the control Vector they're kind of just they're independent param independent variables or they inputs to this set of equations of motion right so what I mean by that is let's just go ahead and consider those two as independent variables I'm going to create an independent variable Vector let's call it Z and I'm just going to stack X on top of U so Z is four elements long and it's just X1 X2 U1 U2 okay now the reason I want to do that is because then I can rewrite my equations of motion and get a nonlinear State space representation and again if you want a refresher on state space representations right we've got a dedicated video talking about that but you can write your equations of motion to look like this okay so it's x dot right which is basically Z1 and Z dot or Z1 and Z2 dot right as you see right here x dot is equal to this ugly nonlinear function okay and this is a vector valued function correct you put in four things you put in these four independent variables Z1 Z2 Z3 and Z4 and this spits out two things it spits out a uh Z1 Dot and a Z2 dot so again think about this this is nothing more than the the vector valued function f that we were talking about earlier okay so I'm going to call this whole thing F and again the picture that goes along with it is this orange picture again the vector valued function f it's nothing more than two scalar functions stacked on top of one another this whole thing takes in four inputs spits out two outputs okay so now we can now start talking about the Jacobian of this F function FB bar right that's all we need to do and again we saw earlier that the Jacobian is basically assessing the sensitivity of all of these outputs to perturbations in these inputs okay so what I can write here is that the change in the output right is the Jacobian matrix times some perturbation in the inputs a Delta Z okay so Delta output is the Jacobian time Delta input again this is just the higher Dimension Vector valued interpretation of a derivative a derivative is basically saying the change in the output is the slope times the change in the input okay that's all this is saying in Matrix form so if I want to write this it's basically um sorry maybe what we should do then is let's calculate this Jacobian J so I'm going to do that over here okay so the Jacobian J as we see it's nothing more than four um independent parameters sorry I missed a parenthesis there okay so it's a 2x4 Matrix of all of these partials okay so all you got to do is just start taking all these partial derivatives this first row is super easy because it's the derivative of F1 with respect to all the independent variables and F1 luckily for us it's super easy it's just Z2 so you end up with just 0 1 0 0 okay now that's not the case for the second row the second row is where all the Dynamics come into the into play okay so it's the derivative of this ugly nasty expression with respect to Z1 Z2 Z3 Z4 okay so this is what you end up with correct this is the Jacobian matrix here and this is your two whoopsie my pen is super dead um I guess this one is dead let's try this blue pen I've been using blue to denote the the the dimensions right this is a 2x4 Matrix okay so here's your Jacobian matrix so now I can plug it into this equation down here which is telling us the change in outputs is the Jacobian times the change in input so it's just this 4x2 Matrix times Delta Z and remember Delta Z is just Delta of all these X1 X2 U1 U2 so that's what I'm writing right here and if you stare at this long enough you can see that okay the X terms here right this multiplies this first 2x two block okay the U terms or your control vectors multiplies this second 2x two block so why don't I go ahead and just I'm going to rewrite this I'm going to break this up okay I'm going to break this up into some 2x two Matrix times your state Vector plus another 2x2 matrix times your control Vector right if you go ahead and if we call this a and this Matrix over here B this starts to look and I I got to be a little bit careful here I'm uh let let me just say this equals a * Delta x + B time Delta U okay now I'm not going to say here that this is x dot okay because that's actually not true we got to be a little bit careful and I'm going to have to handwave a little bit and punt this discussion down to our other video where we will talk about formally linearizing a system because if you stare at this long enough this is the foundation of how we're going to do this what we end up with this should look really familiar right this is sort of like X do is equal to ax + bu U of course we've got these Deltas we're going to have to reconcile a little bit later but at the end of the day like we said this is the foundation of basically your linear OD right it's your linear set of Dynamics and remember where did we start from we started from this function it's ugly it's nonlinear this thing effectively at the end of the day right it's this is describing X do is equal to f of x and U this is your nonlinear OD or set of dynamic so you have an ugly nonlinear system like this you can describe it here and now we can use this Jacobian matrix to basically linearize the system and turn it into and again hold we're not totally turning on so so don't quote me on this right but we we're we're 90% of the way to the x do is equal to a Delta X plus b Delta U right so we're going from nonlinear to linear Dynamics and the Jacobian is the key to making all of this work and again let's just reiterate this for for for to to to Really Drive the point home right all the Jacobian matrix is talking about right what the A and the B Matrix capture here it's the sensitivity of the Dynamics to perturbations in the state and control Vector depending on where you happen to be located and where you're operating in state and control space so if this pendulum is you know caned over like this with a certain set of input conditions right its Dynamics behave differently than if it were say vertically straight up right and the Jacobian matrix captures all of that in a nice graceful elegant fashion okay so I think this is super exciting because yeah this is the key and the foundation for linearizing dynamic systems now I know not all of you are interested Ed in controls and dynamic systems but let's just say this Jacobian matrix idea this is also the foundation for our next video where we're going to start talking about the chain Rule and the chain rule we're going to see is basically uh it's it's a way that we can look at composite functions and try to understand again how pertubations in inputs affect outputs at all of these different layers of this composite function that is going to be the building blocks for a lot of other engineering uh tools for example um in Ai and machine learning there's the concept of a neural network and back propagation and how to train that Network and we're going to see that that is founded on the idea of the chain rule which we are going to show the chain rule is founded on the idea of the Jacobian we just saw the Jacobian is found on the idea of a gradient and the gradient is founded on the idea of a derivative so really all of this is just fancy ways of looking at how the derivative affects a lot of these interesting Eng Engineering Systems so with that being said I think this is probably a great spot to leave it um I hope you enjoyed the video and if so I also hope you'll consider subscribing to the channel um if you scroll a little ways down and click on that subscribe button it really does help me continue making these videos and remember the new videos come out every Monday so I hope we'll be able to catch you at a future discussion and we can all learn something new together so until then I think I'm going to sign off talk to you later bye

Channel: Christopher Lum

Views: 5,194

Keywords: Partial derivative matrix, MIMO system sensitivity, function sensitivity, function derivative

Id: QexBVGVM690

Length: 40min 20sec (2420 seconds)

Published: Mon Feb 05 2024