 And now we move on to chapter 4 in Holland Johnson which is Hermitian and Symmetric matrices. So Hermitian and Symmetric matrices are special matrices which are, I mean I am distinguishing them only in terms of complex versus real, a Hermitian matrix is one where A conjugate transpose equals A and a symmetric matrix is a real valued matrix where A is equal to A transpose. But these arise naturally in a variety of applications. So for example, if you are looking at the, looking at a function f which is defined from some n dimensional real space which I call d to the real line then and suppose it is twice continuously differentiable. We define the Hessian matrix for this f as a matrix h of x which is an n cross n matrix whose entries are h ij of x and each h ij is the second partial derivative of this f with respect to xi and xj. So this is a function of n variables, you take the second partial derivative with respect to xi and xj and this is a matrix that will be in r to the n cross n. And a key property of, this is very important in optimization and especially in optimization differential equations and so on. But a key property of this partial derivatives is that dou square f over dou xi dou xj is equal to dou square f over dou xj dou xi, this is a fundamental property of these partial derivatives for every ij equal to 1, 2 up to n. And as a consequence h is equal to h transpose i e, it is symmetric. Similarly if we look at what is known as the quadratic form, so suppose we are given an a in r to the n cross n and x in r to the n, then consider q of x which could be one example of this function f here, q of x is equal to x transpose ax. Then so note that we can always write a as one half of a plus a transpose plus one half of a minus a transpose, this can always be written. Basically half a transpose cancels with minus half a transpose and half a and half a add up to give you a. So this is always true. So if I substitute for this in here, that means that q of x is equal to half x transpose a plus a transpose x plus half x transpose a minus a transpose times x. And if a is a symmetric matrix, then actually a need not even be symmetric here. So if you just examine this x transpose a minus a transpose x, this is equal to x transpose ax minus x transpose a transpose x, but this is a scalar. So its transpose is equal to itself. So for any scalar, this is equal to scalar. And so the transpose of this is x transpose a transpose x. And so these two are always going to be equal. So this is equal to zero. And so this q of x is actually equal to one half x transpose a plus a transpose times x. Now what that means is that a and one half a plus a transpose generate the same quadratic form. So basically that means this is a very, very crucial point actually. So that means that if you are interested in studying quadratic forms, it is sufficient to study quadratic forms generated by symmetric matrices. So we will be studying quadratic forms generated by symmetric matrices. And finally, a third example is that I'm out of time, so I'll just very briefly mention it in graph theory. We define an adjacency matrix. A graph is defined by a set of vertices and edges. And an edge, so the set of vertices, you can denote it by one to n. So if there are n vertices in the graph, then you can label those vertices as one to n. And edges are pairs of indices, pairs of vertices, i, j, i1, j1, i2, j2, etc. And an edge exists if the two nodes i and j are connected to each other. And we define an adjacency matrix to be a matrix A which contains entries a, i, j where A, i, j equals 1 if there exists an edge between nodes i and j and 0 otherwise. And if the graph is undirected, undirected means the edges don't have a direction associated with them. So you have two nodes, you define an edge like this, not with an arrow going like this. It's an undirected graph, then undirected graphs. So these are three examples where symmetric matrices arise. And so with this background, we will actually start in the next class studying about Hermitian and symmetric matrices. They have lots of very intricate, very beautiful properties. And so we'll cover them.