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# Eigenvectors and eigenspaces for a 3x3 matrix

## Video transcript

in the last video we set out to find the eigenvalues of this 3x3 matrix a and we said look an eigenvalue is any value lambda that satisfies this equation if V is a non zero vector and that says well that means any value lambda that satisfies this equation for V is not a nonzero vector we just a little bit of vector I guess you can call it vector algebra up here to come up with that and review that video if you like and then we determined look the only way that this is going to have a nonzero solution is if this matrix has a non-trivial null space and only non-invertible matrices have a non-trivial null space or only matrices that that have a determinant of 0 have non-trivial null spaces so you do that you got your characteristic polynomial and we were able to solve it and we got our eigen values where lambda is equal to 3 and lambda is equal to minus 3 so now let's do what I consider the more interesting part is actually find out the eigen vectors or the eigen spaces so we can go back to this equation for any eigenvalue this must be true this must be true but this is easier to work with and so this this matrix right here times your eigen vector must be equal to 0 for any given eigen value this matrix right here I just copied and pasted it from above I marked it up with the rule of Sarrus but you could ignore those lines it's just this matrix right here for any lambda lambda times the identity matrix minus a ends up being this so let's take this matrix for each of our lambdas and then solve for our eigenvectors or our eigenspaces so let me take let's take the case of lambda is equal to 3 first so if lambda is equal to 3 this matrix becomes lambda plus 1 is 4 lambda minus 2 is 1 lambda minus 2 is 1 and then all of the other terms stay the same minus 2 minus 2 minus 2 1 minus 2 and 1 and then this times that vector V or our eigen vector V is equal to zero or we could say that the eigenspace for the eigenvalue three is the null space of this matrix which is not this matrix it's it's lambda times identity minus a so the null space of this matrix is the eigen space so all of the values that satisfy this make up the eigenvectors of the eigen space of lambda is equal to three so let's just solve for this so the null space of this guy's is you can just put in reduced row echelon form the null space of this guy is the same thing as the null space of this guy in reduced row echelon form so let's do put it in reduced row echelon form so the first thing i want to do let's let me just do it down here so let me I'll keep my first row the same for now four minus two minus two and let me replace my second row with my second row times two plus my first row so minus two times two plus one is is zero one times two plus minus two is zero one times two plus minus two is zero this row is the same as this row so I'm going to do the same thing minus 2 times 2 plus 4 is 0 1 times 2 plus 2 is 0 and then 1 times 2 plus minus 2 is 0 so the solutions to this equation are the same as the solutions to this equation let me write it like this instead of just writing the vector V let me write it out so V 1 V 2 V 3 are going to be equal to the 0 vector 0 0 just rewriting it slightly different and so these two rows or these two equations give us no information the only one is this row up here which tells us that 4 times V 1 minus 2 times V 2 actually this wasn't complete reduced row echelon form but host enough it's easy for us to work with 4 times V 1 minus 2 times V 2 minus 2 times V 3 is equal to 0 and well let's just divide by 4 now I could have just divided by 4 here which might have made it maybe I skipped a step if you divide by 4 you get V 1 minus 1/2 v2 minus 1/2 v3 is equal to 0 or V 1 is equal to 1/2 V 2 plus 1/2 v3 just added these guys to both sides of the equation or we could say if we say that let's say that V 2 is equal to I don't know it's equal to I'm just going to put some random number a and V 3 is equal to B then we can say and then V 1 would be equal to 1/2 a plus 1/2 B we can say that the eigenspace the eigenspace for lambda is equal to 3 is the set of all vectors v1 v2 v3 that are equal to a times a times v2 is a right so v2 is equal to a times 1 v3 has no ay in it so it's a times 0 nowsave v1 plus B times v2 is just a right v2 has no B in it so it's 0 V 3 is 1 times 0 times a plus 1 times B and then V 1 is 1/2 is 1/2 a plus 1/2 B 1/2 and 1/2 for any a and B I could rino such that a and B are members of the reals just to be a little bit formal about it so that's our eigen any any vector in that may that satisfies this is an eigen vector and they're the eigenvectors that correspond to the eigenvalue lambda is equal 3 so if you apply the matrix transformation to any of these vectors you're just going to scale them up by a by 3 or you could say let me write it this way the eigenspace for lambda is equal to 3 is equal to the span all of the potential linear combinations of this guy in that guy so 1/2 1 0 and and 1/2 0 1 so that's the that's what only one of the eigenspaces that's the one that corresponds to lambda is equal to three let's do the one that corresponds to lambda is equal to minus three so if lambda is equal to minus three I'll do it up here I think I'm gonna paste lambda is equal to minus three this matrix becomes this matrix becomes I'll do the diagonals minus 3 plus 1 is minus 2 minus 3 minus 2 is minus 5 minus 3 minus 2 is minus 5 and all the other things don't change minus 2 minus 2 1 minus 2 minus 2 and 1 and then that times vectors and the eigenspace that corresponds to lambda is equal to minus 3 is going to be equal to zero I'm just applying this equation right here which we just derived from that one over there so what is so we're looking the eigenspace that corresponds to lamp is equal to minus 3 is the null space this matrix right here are all the vectors that satisfy this equation so what is the null space of this is the same thing as a null space of this in reduced row echelon form so let's put it in reduced row echelon form so let's the first thing I want to do I'm going to keep my first row the same I'm going to write a little bit smaller than I normally do because I think I'm going to run out of space so minus 2 minus 2 minus 2 and then actually let me just do it this way I will skip some steps let's just divide the first row by minus 2 so we get 1 1 1 and then let's replace this second row with the second row plus this version of the first row so this guy plus that guy is 0 minus 5 plus minus or let me say it this way let me replace it with let me replace it with the second with the first row minus the second row so minus 2 minus minus 2 is 0 minus 2 minus minus 5 is plus 3 and then minus 2 minus 1 is minus 3 minus 3 and let me do the last row just in a different color for fun and I'll do the same thing under this row minus this row so minus 2 minus minus 2 is 0 minus 2 plus 2 minus 2 minus 1 is minus 3 and then we have minus 2 minus minus 5 so that's minus 2 plus 5 so that is 3 now let me replace now let me in now let me replace and I'll do it in 2 steps so this is 1 1 1 I'll just keep it like that and actually well let me just keep it like that and then let me replace my third row with my third row plus my second row I'll just zero out you sad these terms these all just become 0 that guy got zeroed out and let me take my second row and divide it by 3 so this becomes 0 1-1 and then I'm almost there I'm almost there I'll do it in orange so let me replace my first row with my first row - my second row so this becomes 1 0 and then 1 minus minus 1 is 2 1 minus minus 1 is 2 and then the second row is 0 1 minus 1 and then the last row is 0 0 0 so any V that satisfies this equation will also satisfy this guy that is that this guy's null space is going to be in the null space of that guy in reduced row echelon form so V 1 V 2 V 3 is equal to 0 0 0 let me move this because I've officially run out of space so let me move this lower down where I have some free real estate let me move it down here this this is corresponds to lambda is equal to minus 3 this was lambda is equal to minus 3 just to make us you know it's not related to this stuff right here so what are all of the v1 v2 and v3 that satisfy this so if we say that v3 let's say that v3 is equal to v3 is equal to t if v3 is equal to T then what do we have here we have this tells us that v2 minus v3 is equal to 0 so it tells us that v2 minus v3 right v2 is what 0 times v1 plus v2 minus v3 is equal to 0 or that v2 is equal to v3 which is equal to T that's what that second equation tells us and then the third equation tells us or the top equation tells us v1 times 1 so v1 plus 0 times v2 plus 2 times v3 plus two times v3 is equal to 0 or V 1 is equal to minus 2 V 3 which is equal to minus 2 times T so the eigen space that corresponds to lambda is equal to minus three is equal to the set of all the vectors V 1 V 2 and V 3 where well it's equal to T times T times V 3 is just T right we does V 3 was just T V 2 also just ends up being T so 1 times T and V 1 is minus 2 times t minus 2 times T 4 T is any real number or another way to say it is that the eigenspace for lambda is equal to minus 3 is equal to is equal to the span or lamb I wrote that's really messy for lambda is equal to minus 3 is equal to the span of the vector of the vector minus 2 1 and 1 just like that and looks interesting because if you take this guy and dotted with either of these guys I think you get zeroes that is that definitely the case if you take minus 2 times one half you get a minus 1 there and then you have a plus 1 that's 0 and then minus 2 times one half yeah you dot with either of these guys you get zero so this line is orthogonal to that plane very interesting so let's just graph it just so we have a good visualization of what we're doing so you know we had that 3 by 3 matrix a it represents some transformation in r3 and it has two eigenvalues and each of those have a corresponding eigen space so the eigenspace that corresponds to the eigenvalue 3 is a plane in r3 it's a plane in r3 so this is the eigenspace for lambda is equal to 3 and it's the span of these two vectors right there so you know if I draw maybe they're like that just like that and then the eigenspace for lambda is equal to minus 3 is a line it's a line that's perpendicular to this plane it's a line like that it's the span of this guy maybe if I draw that vector that vector might look something like this it's the span of that guy so what this telling this is this right here is the eigenspace for lambda is equal to minus three so what that tells us just to make sure we're you're inter interpreting our eigen values and eigen spaces correctly is look you give me any eigenvector you give me any vector in this you give me any vector right here let's say that is vector X if I apply the transformation if I'm multiply it by a I'm going to have three times that because it's in the eigenspace for lambda is equal to three so if I were to apply a times X a times X would be just three times that so that would be a times X that's what it tells me this would be true for any of these guys if this was X you took a times X is going to be three times as long now these guys over here if you have some vector in this eigen space that corresponds to lambda is equal to three and you apply the transformation let's say that this is X right there if you took the transformation of X it's going to make it three times longer and in the opposite direction it's still going to be on this line but just in so it's going to go down like this and that would be a times X it'd be the same it'd be three times this length but in the opposite direction because it corresponds to lambda is equal to minus three so anyway we've I think made a great achievement we've not only figured out the eigenvalues for a three by three matrix we now have figured out all of the eigen vectors which are there's infinite number but they represent two eigen spaces that correspond to those two eigenvalues four minus three and three see you in the next video