 Welcome back everyone to our lecture series based upon the textbook linear algebra done openly. As usual I'm your professor today, Dr. Andrew Missildine. These lecture videos that you're viewing right now are going to be based upon section 6.2 from the textbook entitled the Characteristic Polynomial. And so if we pick up the story where we left off last time talking about eigenvalues and eigenvectors, we've seen what an eigenvector and eigenvalue are. We've been able to, we've learned tests that if we have a vector can we check if it's an eigenvector for a matrix or if we have a lambda, a number, can we check to see if it is an eigenvalue for a matrix? And so from this we can calculate eigenspaces. So given an eigenvalue we can find the eigenvectors and all that jazz. But we sort of haven't yet been on it really answer the question how do we find the eigenvalues themselves. We did see an example where we were able to find eigenvalues of triangular matrices. It was actually pretty quick and we're going to kind of revisit that issue right here in this portion of the lecture as well. But we're going to describe the, what's known as the characteristic polynomial of a matrix which gives us a way of finding the eigenvalues of said matrix. So let's, let's talk exactly what we mean by this. We saw previously that a number lambda is an eigenvalue. It's an eigenvalue of the matrix A. This happens if and only if the matrix A minus lambda I is singular. It doesn't have an inverse. It's a singular matrix. Well the matrix A minus lambda I where I capitalize the identity matrix. This will be singular if and only if the determinant of A minus lambda I like so what this only happened when the determinant equals zero in that situation right there. And so the determinant of A minus lambda I equaling zero that gives us a way of calculating whether something is a lambda or whether lambda is an eigenvalue or not. And if we take the perspective where since we don't know, we don't know what lambda is a peory. If we treat this as a variable that is some number to be placed in later. If we treat it like a variable then this determinant of A minus lambda I actually turns out to be a polynomial. And it is a polynomial. It'll be a polynomial of degree n assuming more n, a is an n by n matrix. This will be a degree n polynomial. And this polynomial is known as the characteristic polynomial of A. Now in all of the theory of eigenvalues and eigenvectors and such you usually see this adjective or prefix eigen showing up on front of everything that comes from the German which would translate in this case as characteristic. It seems kind of weird that in the United States here although historically we used a lot of we used to call everything characteristic characteristic values, characteristic vectors, characteristic polynomials, characteristic bases and things like that. And that's because frankly speaking there was some animosity between the USA and Germany back in like the 1940s or something like that where a lot of the powerhouses in algebra lived in Europe particularly in France and Germany at the time. And because of the disruption that occurred from World War II a lot of those scholars for fear for their lives and for other political reasons migrated to the United States and academics in the United States really grew because of sort of that exodus from Europe. And so it's kind of funny if you look at older textbooks on linear algebra you'll often see characteristic values, characteristic vectors, characteristic polynomials but as relationships between the United States and Germany seem to heal over the last several decades you start to see the the German term eigenvalue appearing more and more and more inside of American linear algebra textbook. The one thing that artifact that lived from those days of anger is this characteristic polynomial but as it's this sort of common practice I'm not going to try to reinvent the wheel in this situation. This this polynomial the determinant of a minus lambda i is called the characteristic polynomial of the of the matrix here and the roots the roots of this characteristic polynomial are exactly the eigenvalues of a. And so that's why we're going to care about this characteristic polynomial is because it helps us find the eigenvalues. Now it is a polynomial the eigenvalues will show up in this polynomial with multiplicity and that multiplicity we commonly refer to as the algebraic multiplicity. So the algebraic multiplicity says how often does an eigenvalue show up inside of a polynomial that the characteristic polynomial and the geometric multiplicity which we talked about before it counts the dimension of the associated eigenspace. And so one thing I want to kind of point out to you here is that if we take n to equal the algebraic multiplicity of an eigenvalue and we take m to be the geometric the geometric multiplicity remember geometric is measuring the dimension of the of the null space a geometric object the algebraic multiplicity counts how often something shows up in the characteristic polynomial and algebraic thing. What is always true is that the geometric multiplicity of an eigenvalue will always be greater than equal to one because if it was less than one it wouldn't be an eigenvalue. But it's also always less than or equal to n that is the algebraic multiplicity always is is an upper bound to the geometric multiplicity but there are situations for which there is a gap between the geometric and algebraic matrix multiplicity I should say. We'll talk about these more in the future when exactly is that bound obtained. But I want to do some examples some quick examples of computing characteristic polynomials and so these you'll see in front of you two triangular matrices which you might remember from before we calculated their eigenvalues before and I'll kind of compare what we did there with what we're doing right now we want to calculate the characteristic polynomial of this matrix so we take the matrix a and we subtract from it lambda minus i I do want to mention that some authors actually define the determinant to be sorry the term they define the characteristic polynomial to be determinant of lambda i minus a it really kind of comes down to a matter of flavor in terms of the polynomial it just times the polynomial by negative one and so it has no effect to the eigenvalues whatsoever so we'll just stick with this a minus lambda i approach here this matrix is triangular even when we subtract lambda i from it since a was triangular a minus lambda i is also going to be triangular so the determinant is just the product of the diagonals so you're going to get three minus lambda you get negative lambda and then you get two minus lambda like so and so if we proceed to start multiplying this thing out we can distribute this negative lambda onto the second bit right there and so the three minus lambda will be kept where it was we then get negative two lambda plus lambda squared and then if we foil out this last part here we're going to get negative six lambda plus three lambda squared plus two lambda squared and then minus lambda cubed combining like terms we have a negative lambda cubed plus five lambda squared minus six lambda and so this right here gives us the characteristic polynomial of the matrix right here it's this degree three polynomial right there and honestly speaking though when it comes to the characteristic polynomial we are going to prefer to have it factored because if it's factored when we set it equal to zero we can very quickly determine what the roots of this thing are by the zero product property we either get that lambda was three lambda was zero or lambda was two and if I remind you of that example we did earlier three zero two those are exactly the diagonal entries of the matrix and so the eigenvalues three zero and two coincide exactly with what we saw before so for a triangular matrix we don't necessarily need the characteristic polynomial to find the eigenvalues but I want to show you that these things do in fact agree with each other as another example let's take our other friend B which is this triangular matrix right here as it's triangular B minus lambda I will likewise be triangular so the characteristic polynomial becomes four minus lambda times one minus lambda times four minus lambda you'll notice that four minus lambda shows up twice so in factored form we might write four minus lambda squared times one minus lambda and so this is suggesting that the algebraic multiplicity of one one right here so we get lambda equals one is the first eigenvalue and then that the algebraic multiplicity of one is itself one but then when you look at the eigenvalue four its algebraic multiplicity is a two so those things are a little bit different there you're just looking at the power in this characteristic polynomial if we were desirous to multiply out we will do so we'll take four minus lambda quantity squared that gives us 16 minus eight lambda plus lambda squared times that by one minus lambda and so when we multiply that out we get 16 minus eight lambda plus lambda squared and then continuing on we're going to get a negative 16 lambda plus eight lambda squared and then minus lambda cubed and so in the end we end up with 16 minus 24 lambda plus nine lambda squared minus lambda cubed and so this right here gives us the characteristic polynomial of that matrix B and like I said you can do you can do the determinant of B minus lambda I you could also do the determinant of lambda I minus B I really don't have a preference one way the other feel free to do either one of these things this approach right here always has the advantage that the leading coefficient will be one because you'll notice in this example it turned out to be negative which isn't such a big deal we have to end up factoring the thing anyways and if you pick one convention versus the other yours times by negative one all at the end anyways so this gives us some examples of computing some characteristic polynomials take a look at the next video for which we're going to take a look at an example where the matrix is not triangular and how do you find the characteristic polynomial there see ya