Lec 2 MIT 18.085 Computational Science and Engineering I

two and the topics are applications in one dimension uh and uh this is what's new this is the key part that's new here so now we're dealing with the question of finding the matrices finding the equations where do they come from and this is the going to be the key to that so again we we will end up with one of these symmetric products an A and an a transpose it happens that the a goes on the right and the a transpose on the left but the point is one is the transpose of the other and uh the product is of course square and symmetric okay but now where what's the application and these are going to be uh the matrices that come up uh if I create the right application and the discussion in the text is in section 1.4 I there're going to be examples with masses and springs because that's the most basic problem of mechanics so I know that your interests go all over the field of Applied Mathematics and uh lots of it so mechanics uh are sort of the natural topic when I'm lecturing to uh graduate students say in this is close to the MIT course 18.085 18 for math 085 is the course that's taken by a lot of mechanical engineers aeronautical engineers and some electrical engineers but of course as you know the interests in electrical engineering and in estimate well we'll see least squares and estimation problem s already next Thursday so we'll try to cover a whole range this so here's the application that I claim is going to lead to that matrix it's going to be just a line so onedimensional of so a spring and then a mass then a spring and a mass and a spring oh sorry I'm using more space than I should and a spring because I need three masses to get a 3X3 Matrix and it's going to be fixed at the bottom so I've got four Springs with three masses okay that's that this goes with k k can I write down right while we're looking at it just what the other ones are the next one the one that goes with t oh H damn uh uh I've I've Chang I've put the boundary condition at the top ah how can I make a tower of you see my problem I I I I yes it's like it's like a well okay Skyhook a skyh all right well I better Dre the one that I'm sort of expecting which is to have this same deal but then what's going to be different at the at the boundary at the at the lower boundary it's free at the thing ends there so it's got three masses now only three Springs and um I better fix the Matrix because it's not because that that one has the has the has some has the free this this is associated with a free boundary and I want the free boundary at the bottom so can I put the one here and the two back up there of course I still haven't created these matrices that's going to be my main point and maybe I'll call it H3 for hanging is that all right okay so we're now we're and now what is this one going to be anybody want to make a guess here this is a crazy way to lecture sorry that's on tape but it's much better if you know what's coming uh then uh you're you're right totally with me okay any guess on this one oh no I don't want that two how did that two get there okay yeah I just wanted to be back to that one which was remember that was the singular one and exactly that's got a mass spring Mass spring and mass but nobody's holding it there it's not it's to it's free free you could say so not surprising it's singular it's like WEA I'm sorry to hear it uh uh do you remember what was the singular what was the vector in the null space that we identified for this one 111 and and and what does that one one represent physically the one one one represent the displacements of the masses so what I'm saying is I could take the whole thing and shift it by any amount so it's it's like uh totally arbitrary constant in there it's it's almost like calculus where you do those integrals and you always have to remember plus C or you lose a point or something okay right okay well this is the this is the plus C in in Matrix in linear algebra right okay so this no wonder it's singular right I mean there's no way we could determine the displacements because the whole thing could could rigidly shift so that's a rigid motion and uh and nothing's preventing it whereas here those are perfectly statically stable uh structures right I mean if I If gravity acts so these are masses M1 M2 and M3 and gravity and similarly there and gravity is pulling them down then it's going to stretch those Springs it'll stretch these Springs well it'll actually compress that spring I guess won't it okay so we need to we need to think okay what are the unknowns what is it we're what is it we're trying to solve for in this Mass spring situation what's the setup okay okay now so the here's the structure I'm I'm interested in how how much the masses what's the displacements I'm interested in the three displacements so let me write those down so let me call them U1 U2 and u3 those will be the displacements of the masses of the three masses okay that's a principle I want to know I mean that's that that's something to to that's an unknown U for unknown okay another unknown is the forces in the springs and let me call those uh W so that'll be now there'll be one for every spring W1 W2 this is this is n three is n here I'll I'll try to be consistent that n is the number of displacement unknowns uh here these These are associated with springs well here I've got four Springs so let me stay with this picture and and emphasize that there are four Springs here this four is what I'll call M so here M would be three here M would be two and that's an absolute signal of trouble when m is smaller than n so this is this I'll just write that down again m is only two here n is three I have M less than n trouble okay but so so what were these this four was M uh and these are the the internal forces in the spring right if if the if the mass If gravity acts so we have forces pulling downward on the masses the Springs react this is the reaction force from the Springs the spring force can I call it that the forces is the tension let's let me call them tension spring tension and of course I have to say a word about Signs Now I I'll tension will mean a a positive W would be tension so tension is a w positive but of course like that last spring is going to be in compression or compression and compression I'll is the it would be a negative W okay so those are my primary unknowns the U's and the W's I'm I'm and and now what's the external Force so the external force is gravity so I have external forces fub1 F2 F3 and those are external forces on the masses Force on the mass gravity in fact in this simple example okay okay so what's my goal here I'm given let's see what information do I need to I would like to find these displacements and find these forces in the springs these are the unknowns this is a known quantity here there's one other there's a fourth box so what I'm describing here is the fundamental framework for equilibrium problems equilibrium problems mean steady state these I'm not these Springs are not bouncing right now they're they're the mass pulls them down they settle down I want to know what those displacements are what the spring forces are and the in between quantity I want is there'll be M of them because they're in the springs E1 E2 E3 E4 if there are four Springs and that e stands for elongations stretching of the spring how much the spring stretch of the Springs okay so I'm emphasizing these four quantities and now I want to get some equations up how are these connected see ultimately this is what I'm given this is the right hand side of my equations and I'm interested said what's the left hand side now what what's the what's the what equals the external Force okay and here's my plan my plan and this is I've just found that this is a framework that I see over and over and over again in lectures in in lectures of the widest variety statistics chemical engineering wherever I just wait for this uh structure to appear and uh does so the the point is that these U's so let me copy it again here the U's there'll be a matrix a that connects the U's to the E there' be a matrix C that connects the e to the W's and then there'll be a matrix that connects the W's to the fs so this is like vector notation if I put arrows over them something you realize those are vectors and anybody want to guess what that Matrix is it's a transpose it happens that so let's figure out these matrices what's the matric what's the connection if I know the displacements how much do the Springs how far do they stretch so I'm I'm interested now in this connection U equal e equal Au how much do the spring stretch so that's Matrix language it's a vector of four e's a vector of three U's I I'm working I'm working always with this this example okay so ready for ready for help here how much does that spring stretch if if that one's fixed so you could say u0 is zero it's fixed this one drops by U1 how much does a spring stretch by E1 will be U1 right just that so that we got a simple equation there the the the the so I'll just write that down the first spring stretches by U1 okay that's because the top of it was fixed now let's take a typical spring the second one what's the stretching in the second spring if this goes down by U1 and this goes down by U2 how much does the spring stretch it's going to be a difference it's going to be a U2 minus U1 because if they won't if the masses go down the same amount then the spring won't stretch right it's the difference that matters so this is U2 minus U1 one and what about the third spring how much does it stretch if the displacements are U1 U2 u3 u3 minus U2 because that's the different that difference controls the third spring and now the fourth spring is the stretch in the fourth spring is is minus yeah they're good to see that minus minus u3 because you see that the stretching when this comes down it's not stretched then if if u3 is positive if that mass comes down then the stretching is negative the elongation is actually a shortening and it's and this this end is fixed so so it's minus u3 right good good if for this example everything would look the same but there wouldn't be any fourth spring so that fourth equation would go okay now I'm ready to say what's the Matrix that's what's the Matrix a having written out the the four equations let's write it let's write them in Matrix form okay so so the four equations are E1 E2 E3 and E4 is some Matrix times U1 U2 u3 and what's the what's the first row of the Matrix one Z 0 in order to pick up that U1 I want one Z 0 the second row I want minus a U1 plus the U2 right the third row is minus the U2 plus the u3 I'm looking there I'm just copying that out into the third row and then the fourth row is just going to be well you see this pattern so the fourth row was minus u3 that that's our Matrix a our Matrix a is rectangular first so let's notice the key some key facts about a it's a rectangular Matrix that's we expect that because it's connecting it's giving a connection between n masses and M Springs so it's two different quantities two different numbers it's rectangular uh what kind how would I describe that Matrix in words I would call that it's taking differences I mean here we we took differences so I would call that a first difference matrix it's taking the difference between uh U2 when you w so that that's a difference Matrix and it's built in the boundary conditions cuz the Matrix you know you if tlets were here he'd say there should be a minus one there right or and there should be another one here but what happened to those so right the The Matrix kind of got chopped off that that that diagonal of ones ended suddenly because and and this diagonal of minus ones ended early because the the boundary conditions came in you ran into the boundary right that's right exactly so the Matrix a includes the boundary tells us something about the boundary conditions on you it's it's the boundary conditions on you are reflected there okay so that's that's the connection that's this first step uh I suppose I'm want to say this that we're creating we're doing applied math now creating the matrices but notice how simple they are if we see so we don't have to be like uh um experts in elasticity theory for this example but nevertheless what's the structure of this example the framework is exactly repeated in the much tougher problems of two-dimensional or three-dimensional last icity it's just we have more complicated matrices but the structure is what we can keep hold of okay now this step what's the connection between the the the the distance of spring stretches and the force they're proportional and who's everybody all the old-timers got their names on laws here uh which it's too late for the rest of us but so what what what so that's the that's this here is the sort of this the physics is in this step the the constitutive law is in that Matrix C so I'm using that letter C for constitutive law for example and who who's who cooked up that law hook hook right so this is going to be Hook's law and what is Hook's law it's Spring by Spring right it says that in the first spring the force and the distance stretched are proportional and the proportionality constant is the elastic constant for that spring so so this step is going to be really easy it's just going to be W1 is that first spring constant whatever it is times E1 that's that's his hooks law so so this I'm just going to have four hooks laws here for the four Springs W2 is C2 E2 W3 W4 is C4 E4 and this of course the same for W3 so that's I have a I have a simple relationship there if it's true well so is hooks law true approximately true we'll give it to them right okay for uh but this is a this is a key Point here that we're going to do linear problems linear equilibrium problems and and this is where the linearity comes is comes in that that Hook's law says that approximately the elongation and the force that's needed are proportional and of course if the if the elongation is very large that Pro that law won't any longer hold this is these linear laws are in the in the framework of small displacements and small forces and uh fortunately lots of uh applications uh are in that in that regime and we can deal with Hook's law so what's the Matrix then so now I want to write that as a matrix problem uh I had e equal Au with these differences and now I've repeated e equal Au there so I don't need it here anymore because it's all in that Matrix okay now I'm ready for what's this W equal C and tell me the Matrix C so this is W is C so what uh so the four W's forgive me for doing this simple one again what's the Matrix then that multiplies I'll put it right next to the e um so what's the Matrix in those four equations it's diagonal it just has W1 is C1 E1 and it has C2 then it has C3 then it has C4 okay so there's our Matrix C that's diagonal okay good and now finally you see this roundabout way I I really want to know the this connection how are the forces in the dis placements related but the the easy way to get to it is around the loop connect the U's to the to how far the stretching is to the forces that it took and now what's this what's going to be the final equation what's the physical law that connects those internal forces to these external forces they have to balance right well it's it's it's certainly the most fundamental if we're if we're interested in finding equations this equation of force balance conservation of something conservation of charge if we were doing uh currents uh conservation of matter if we're doing fluid flow that that there's always a key equation is something uh is balanced and here what is what's the balance here the balance is we're in equilibrium so it means that what's the balance at Mass one what's the balance equation so so now I'm ready to tackle uh the balance equation at Mass one so let me draw what's what's balancing there so gravity so what are what forces are pulling on that on that little spot the Spring's pulling up if it's in ttention the second spring is pulling down and gravity is pulling down right so the three pieces are let's see so what's pulling up was W1 and what's pulling down was W2 right and also the gravity so M1 G if we're if we're really uh careful we should put in the so that's the balance there okay okay and now a similar balance there and a similar balance there is that right so of course I want to write those as I want to get this in a matrix form so this is my F1 it's the mass times the gravity gravitational constant so that's the external force from gravity and of course I'll put this over on the the W's on the their side of the equation so it'll bring W2 over with a minus sign so here's how I want to want to look at that first equation W1 minus W2 we please be patient because we're there uh the the uh first equation is W1 minus W2 is fub1 the second equation is what W2 minus just by by uh uh follow the pattern uh the second equation will be W2 minus W3 is F2 in the end we're going to get to an equation that connects the fs to the W's and I want to know what's that Matrix and we've already guessed what it might be but let's see it happen all right let let me let me let me do it because this is like the crucial moment in the lecture is to see that Matrix come out okay so so let me copy W1 minus W2 is F1 W2 - W3 is F2 and W3 - W4 is F3 right those are the those are the balance equations balance the forces balance that's that's the equations of equilibrium steady state crucial equations and and what's the Matrix that's multiplying W I'll I'll write that Matrix beside this W again if you'll allow me and finally we have the fs so so can I can you pull out the Matrix that let me make a space for a matrix here to multiply the W and just tell me what that Matrix is what's the first Row 1 - one 0 0 Z what's the second row zero because there's no W1 in it a one a minus1 and there's no W4 okay and the third row is no W1 no W2 a W3 and a negative W4 okay there's the Matrix then that's the Matrix and now what's the what's the key point about that Matrix what what name shall we give it a transpose because it is exactly the transpose of this a so that so that without uh I mean nature produced a transpose there we didn't we just sort of watched it happen and so this is then the three the three equations that take us around the framework and there it's completely typical so we'll see other examples in which uh uh in the middle comes the the law of physics or or engineering or statistics whatever uh whatever sort of external law comes in these laws somehow produce the Matrix a and its transpose and they come sort of more from the geometry how things are connected so this Matrix a kind of reflects the connections of how how is mass one connected to mass two and so forth and trans and it's transposed so that's uh that's the situation now I'm ready to put them all together okay so if I put these equations together so now I want to now I want to take the shortcut what's the connection between F and U well f is a transpose W maybe I can uh take that spring out and go here now okay I'm putting those together f is a transpose W but W is C but the E is au so there's the there's the we've we've done elimination we've eliminated W and E and we've we're left with the equation we wanted that connects the displacements that the the the output from the forces which are the input and this is called the stiffness Matrix that's the stiffness Matrix and the letter K is the standard letter for that so that's why we're using K so K is a transpose CA and that's the what what we expected to happen okay good and maybe I just say where um so this is like one dimensional Springs and masses so our equation is Ku equal F so this is that's our final equation then KU equal F and in the in the static case nothing moving what's the most important equation in Dynamics the most important equation in Dynamics has the KU and the F but there's motion so Newton's law says acceleration mass times acceleration comes in so when we allow things to move when we look at Dynamic problems there's going to be a term m a mass times acceleration I'm not ready for that today but that you I think everybody would agree is that that mu prime plus kual f is the central equation of uh uh of Dynamics and so for example the finite element method some of you may be experts in using it so the that's sort of become the most powerful widespread method for doing structural problems structural analysis if you have a bridge or a tower or a building uh or many many other structures you would use the finite element method to estimate the displacements and the stresses and be sure that your materials could stand them and so what so this is exactly what happens in the finite element method this this Matrix K has to get assembled the the first step in the finite element method as setting up these little pieces of your structure looking at the connections they'll go into a matrix a looking at the elastic constants that'll go into Matrix C and then assembling that product a transpose CA so when I come to find that elements later this spring what I'll be doing is exactly this uh on a on a much larger more complicated problem okay okay so now uh let's do this multiplication on uh and easy problem on on our problem first okay so I I'm interested now to do that multiplication could I start with the case when C is the identity Matrix so what does that mean in in my problem c will be the identity Matrix when all the spring constants are one so let's suppose all the Springs are the same that's reasonable and we can normalize those constants to be one so let me start with a transpose a before I bring the C part into the picture okay so what is a transpose a so I can erase this because I've it's copied now in the in the in the a transpose okay so everybody's I mean so like this framework don't I won't erase that because that's that's that's what this lecture is illustrating that framework U to e to W to F and these are the matrices that do it now could you multiply so I think of this as a first difference Matrix and I guess it's transpose is somehow also a first difference or maybe minus a first difference anyway here's a Tren can you multiply a transpose by a without my writing them right next to each other let me I'll maybe I'll take these vectors out of the middle so we've got a chance okay now I'm thinking C is the identity but I'll leave it there okay oh yeah now you can do it okay so C is the identity so it's not doing anything I want to multiply a transposed by a uh what do I get in the one one entry two what do I get well tell me the whole first row well tell me the shape of the Matrix first of all this is 3x4 and of course its transpose is 4x3 so what's the shape of the product 3x3 right and now we can identify what it is so what's the first row then I want one of these so what do what what do you want me to do it better be our friend it is a transpose a is this guy now we now we're now going to recognize this as K3 so I've kept the letter K and the number three and I had to specialize to see being the identity Matrix but but now are you okay do you want to like take any special cases on the diagonal I have say second row times second column and that gives me a two off the diagonal I have like second row times third column and the zero the one the minus one is the is what shows up in the off the diagonal okay so a transpose a is is the K that I've written above now should we tackle it with the c in there I think so just to see what uh what it looks like in the physics when we've got the physics in see we we sort of took the physics out when we let C be the identity now put that back and multiply those three matrices ah can we do that so now I'm actually asking you what this multiplication I'm actually asking you to do the finite element method Assemble assemble the stiffness Matrix uh how are you going to do that multiplication let's see uh well fortunately it's diagonal matrix so that'll be that let's do that times that first okay do you mind helping me through this one so this times this will put a C1 there and this will be a c2s and this will be c3s and this will be C4 right because I just multiply the rows by the C's okay now that times that I'm ready for the for the for this for this triple product but where am I going to write it um erase the C we don't need erase the C huh and then put the answer in the middle all right all right that'll make that's got a lot of space and we okay all right so was that okay in the auditorium C just disappeared but it that was because it I've multiplied by the C there okay now now now I want to do a transpose CA okay so that Matrix times that Matrix can you tell me what it finally is what's the shape going to be 3 by3 okay so I can't get out of hand okay the first what so what do I see in the first row do I see C1 plus C2 and then in the next row what do I I mean the next next to it it's is minus C2 I think and then zero is that all right and I know it's symmetric so that that times this will give me the minus C2 all right now what about on the diagonal row two * column two what do I get C2 plus C3 thanks C2 plus notice these positive numbers showing up on the diagonal and then the other side of the diagonal is minus C3 and everybody sort of checking immediately if all the C's are one are we getting the right answer and we are right if all the C's were one we'd have the minus one 2 minus one okay and now what about this this row times this column gives me a zero and and sort of can you tell me why I'm getting a zero there from the from the from the physics here why is there a zero in the 31 and the 13 position there's no effect because there's no yeah because Mass one and mass three aren't connected if I added another spring that would produce something there it wouldn't change the framework it would just it would change a a little bit it wouldn't be quite so it wouldn't be so perfect a problem but it okay so that's why that zero is there because there's no connection between mass one and three directly okay and then the next to that would must be a minus C3 and down here is C3 + C4 okay yeah that's the that's the form of the of the stiffness Matrix and what I now what what is I what is it my what's my claim what kind of what are the properties of this matrix it's Tri diagonal still right it's still symmetric so are all our properties still true it's still symmetric that's because a transpose CA comes out symmetric we checked that for LDL transpose and same here it's still triagonal still triagonal because um because there's no connection because because our system is on a line Tri diagonal matrices are reflecting one 1D problems this is a we're only onedimensional here each person connected to its two neighbors that's what produces triagonal matrices and that's why they're so important so it's local effects neighbor only neighbors only that produce these banded matrices okay what other properties is it is it positive is it Inver now I'm coming to the The Not So Clear part is that Matrix invertible you can say wait a minute how I mean here's something well I guess Mathematica is better than mat lab but physically it should be which and that's important physically this system should be well determined if I tell you the the masses that should determine the displacements so it is it is invertible of course I'm going to ask is it positive definite okay and that's where I want to use you use an idea here okay so what does positive definite mean I could check all the pivots well the first pivot I'm happy about but uh the next pivot might not be so easy I don't know exact yep I don't really want to know what it is or even the det okay the igen values I don't want to know so I'm left with the testing whether X transpose a transpose c a x is that positive right right okay and that was really the definition now can you see so this is this is not using the pivots because we ar it's those would look messy it's not using the igen values they would be even worse it's using this fact that if I take this Matrix which which is a transpose CA I multiply by X on one side and X transpose on the other does it come out positive okay how anybody any suggestion on how I order I wan to I want to get that yeah maybe the way to think is aose right right right so this will be yeah this will be an energy yeah this will be an energy yeah ax is a displacement actually this is the energy in the springs that's the energy in the springs because that's the displacement e of the spring there's the spring constant and there's the E so actually yeah actually that's the the key isn't it do you realize that can I just say ax is the same as oh well there I've used you can I shall I shall I so like yeah X is any Vector yeah it's not the particular Vector oh yeah okay here's my idea I I I like this combination ax why do I like that combination ax because over there is it's transpose it's really C let me call that thing e whatever X could be anything so e is some ve e is just ax then this is e and what's that it's e transpose L waited that's right I've got I've got a positive quantity here let's just see so ax I just gave a name to e does it have to be that letter no I'll accept another letter oh you're you're at liberty change that letter why would be fine okay right yeah maybe maybe would you rather why okay so all right so ax I'm just calling e i I don't know what ax is but it's an a vector I'll just call it e what the point that I do know is that this Vector is its transpose so that this thing simplifies to e transpose C but now how do I know that that's positive well what is it I mean we could figure that one out now that's that's this diagonal matrix which I which so this is this is the E times the times this diagonal matrix C1 down to C4 times the E E1 to E4 I can I mean this is multiplication I can do this is E1 over to E4 this is this is important now and um uh do can you do that multiplication and what do you get you just get a sum of squares you get a waited by the C's you get a C1 E1 squared and a C2 E2 squar and so on right it's positive it's positive so that's the we have actually shown that this Matrix is positive definite by by looking at it the right way we that Matrix was positive definite because the rule is is it giving us an energy that's always positive and believe me that's the in finite elements that's the key to have you have you got a good finite element you you know you're in trouble if the energy can never be zero here it's never zero it's it's definitely positive as long as e isn't zero Vector which of course we have to exclude okay so uh I guess uh my point then is that this little uh manipulation in linear algebra has verified the crucial property of the Matrix that it's going to be positive definite and in particular it's going to be invertible it's going to have positive igen values it's going to have positive pivots uh the elimination algorithm will go perfectly and the igen values will'll use in the dynamic problem okay yes all right this is the closing moment right so that was a good question may I repeat it I guess I have to repeat it right since it's not okay the question is could e have been zero you see we're we're right if e was Zero then what would this result come out to be zero and that's like the danger right so so let me let me write this down as a state statement then okay uh so so this this this a transpose CA is positive definite if now what do we what do we need we we need to know that C is good and of course C is very good in our problem C is certainly positive definite so if C is positive definite and it's diagonal even so no problem but and something about a has to come into it that that's your your question and it's absolutely right because if we don't control a well if we don't control a then we could have had this example yeah oh yeah maybe that's the way to way to make this final point point that that our we've been creating this Matrix from this structure fixed fixed now from the fixed free one the same structure the same plan the same a transpose CA but would have been 3x3 would have produced this guy what would have produced this one this is also an a transpose CA but it's not positive definite and my question is what went wrong what went wrong in this in the B one so I let's figure out what the Matrix a would have been and we'll see that it's that it's uh got a problem so I'm now turning to this singular case to see that uh a problem could have Arisen so so so we're at the last minute or two of the of the lecture can you create the Matrix a for that you remember what this structure was it was a mass spring Mass spring mass and nothing holding it okay so the Matrix a for this one it's m byn always now there are only two Springs and three masses so it's 2x3 and can can we review this what was the where did a come from it came from what's the stretching in this spring it's was minus one of the top displacement and one and zero do you remember that for the first row and the stretching in this spring if this goes down by U2 and this down by u3 we have a minus U2 and a plus u3 okay somehow that Matrix isn't enough because if isn't enough to make it positive definite because if I compute a transpose C with that Matrix I'll get this answer and it's singular it's not positive definite it's only semi-definite in other words it will still be true even in this case my little e transpose C which would still this is still e transpose C and it will still it could never be negative it will still be greater than or equal to zero but now we have the possibility that it equals zero and and why is that because this e could be the zero Vector remember e is ax and if x who's the guilty X right exactly what's what's the X that's going to going to throw us off it's the one one one vector right right if this x if the x is the 111 then when I multiply that a by the 11 one I get the zero Vector so that's what I have to prevent I have to prevent if C is positive definite and the and and some what I have to say about a is um only only if x equals z should I say it that way and ax is zero only if x is zero that the only way to get zero stretches is not to move at all that's telling me that I have a nice non- singular problem this I can get zero stretches and move because I can move the whole thing but these if I move anything something's going to stretch and E will be not zero and I'm positive definite I I didn't know I was going to get to that thanks to your question that uh key point there that the test for positive definiteness is well C is always positive the Hook's law would be crazy if we had uh a negative spring constant but we also the connection Matrix has to has to have this additional property that that rules out instability see this is this example is up here because it reminds us of the possibility that we haven't got enough connections nothing's nothing's fixed there and the and the displacements could be large with zero stretching of the Springs so that would that's then the final bit of linear algebra so you've you've seen this afternoon then quite a bit of linear algebra and the Central Central Construction was always that a transpose CA okay and that's section 1.4 and my lectures next time would be uh statistical estimation and leas squares which is the key on rectangular matrices and networks uh network problems which so we're seeing like networks everywhere in uh in modern engineering so those are the two lectures for next time thank you