 As always, whenever you give a mathematician a new object, the first question they ask is, what can we do? And so the first thing we do is we define what's called the eigen space. So given any eigen value, the eigen space is going to be the set of linear combinations of the corresponding eigen vectors. Now it's a set of linear combinations. We have vector addition, scalar multiplication defined, so one question we might wonder about is going to be, does this set of linear combinations of the eigen vectors for lambda does this form a vector space? And it should actually be fairly easy to verify that the eigen space is in fact a vector space. And we can go through our checklist of what we need for a vector space and see that the eigen space does actually fit all of those requirements. It's a proof that you should be able to do it in a few minutes. So you should take a few minutes aside and convince yourself, prove that that actually is the case. Now that we know that the eigen space is a vector space, is there anything else we can say about it? Well let's think about that. So let's take a look at how we found those eigen values in the first place, the eigen values, for any matrix. Well we solved an nth degree polynomial equation formed when we set the determinant of a particular matrix equal to zero. Again, the eigen problem always has a trivial solution, we're interested in non-trivial solutions and that's going to occur when we get a coefficient matrix with determinant zero. So now this is a polynomial equation, its degree is n and so we know from algebra that such an equation will typically have n roots, some of which might be complex numbers and importantly some of these roots might actually be equal to other roots. And when we were looking at algebra we talked about this as the algebraic multiplicity of a root. And so we can talk about the algebraic multiplicity of an eigen value which is the solution to one of these equations. So for example let's take the matrix 313246002, let's find the eigen values of a and so again the eigen values are going to be the solution to determinant equal to zero, so I'll form the corresponding matrix, I'll subtract lambda down the diagonal, I'll get my new matrix, I'll find the determinant, after all the dust settles my determinant is going to give me solutions lambda equals 2 twice and lambda equals 5. And so we say that lambda equals 2 is an eigen value of algebraic multiplicity 2, lambda equals 5 is an eigen value of algebraic multiplicity 1. Now you might wonder why we specify algebraic multiplicity in both cases because when we were talking about solutions to equations we never had to use that term. We just said it's a solution with multiplicity 2, a solution with multiplicity 1. And the reason that that's important is remember that the eigen value is only half the solution to the eigen problem. This eigen value is going to be used to find the eigen vectors for this matrix. And it's conceivable that the eigen vectors there may be more than one eigen vector for a given eigen value. And so we can also talk about the geometric multiplicity of the eigen values. Now for that we actually have to find the eigen vectors, so let's go ahead and find that. So lambda equals 2 is an eigen value of algebraic multiplicity 2 of our matrix and the corresponding eigen vector is going to be a solution matrix times vector equals 2 times the vector. And so I can set that system of equations up relatively easily and I can then row reduce the corresponding coefficient matrix and I get my row reduction here. I can parameterize my solutions, s, t minus s minus 3t, and so our eigen vector, first component, second component, third component, and I can split that. I have two parameters so I'll split that into two actual eigen vectors, negative 1, 0, 1, negative 3, 1, 0. And my eigen vector is going to be any linear combination of those two basis eigen vectors. Now since the eigen space for lambda equals 2 can be expressed as a linear combination of these two vectors, we'll say that the eigen value has geometric multiplicity 2. And this corresponds to the number of eigen vectors I get for that particular eigen value. What about lambda equals 5? Well, we found out that this was an eigen value of algebraic multiplicity 1 and so my corresponding eigen vector is going to be a solution to Ax equals 5 times y vector x. And this corresponds to finding a solution to the system for second and third components are 5 times whatever they were. And I'll rewrite the system as a homogeneous system of equations, I'll reduce the corresponding coefficient matrix and after all the dust settles I end up with this and I get my parameterization, x3 is 0, that's what I actually get from this third row. x2 is going to be my parameter and x1 is going to also give me that value and so I get this parameterization of all of my eigen vectors. And there's only one eigen vector so our eigen space has geometric multiplicity 1.