In this segment, we'll discuss what are the eigenvalues and eigenvectors of a matrix, and how to find them. First we need to define the determinant of a matrix. A determinant is a scalar property of square matrices and is denoted by the shorthand det or two vertical bars. There's a physical interpretation of the determinant. Think of the rows of an n x n matrix as n vectors in Rn space. The determinant represents the space contained by these vectors. We'll see more specifically what this means in the next few examples. In this course we will be working with 2x2 or 3x3 matrices. Consider a general 2x2 matrix defined by the variables a, b, c, and d. The determinant of this matrix can be found with the following formula. Consider each row of this matrix as a vector in R2, or in the 2D plane. These vectors define a parallelogram. The magnitude of the determinant of this matrix represents the area of this parallelogram. Consider the following matrix. The determinant can be found by directly using the formula we saw on the last slide. Now consider a 3x3 matrix. The determinant of this matrix can be found using the following formula. If we consider each row of the matrix as a vector in R3 space, these vectors define a parallelepiped. The determinant of A is the volume of this parallelepiped. Consider the following matrix. Again, we can find the determinant directly using the given formula. Knowing how to calculate a matrix determinant allows us to find the eigenvalues and eigenvectors of a square matrix. We can think of a matrix as a transformation. That is, when we multiply a matrix A by a vector x, we transform the vector x into a new vector y. This transformed vector y could possibly have a different magnitude and/or a different direction as the original vector x. Eigenvectors are vectors that do not change direction when multiplied by the matrix. However, these vectors can still change in length. Eigenvalues are scalar values representing how much each eigenvector changes in length when multiplied by the matrix. There will be an eigenvalue corresponding to each eigenvector of a matrix. However, multiple eigenvectors can have the same eigenvalue. Mathematically, this is represented by the equation Av equals lambda v. Here A is a matrix, v is an eigenvector, and lambda is its corresponding eigenvalue. Consider the two-dimensional vectors a and b shown here. Suppose we transform these vectors by the matrix A and get the following result. Here we see that b is transformed into a vector that points in a different direction and has a different magnitude. Thus, b is not an eigenvector. However, the vector a remains pointed in the same direction. This means that a is an eigenvector. If a does not change in magnitude, the corresponding eigenvalue is 1. Given an arbitrary matrix, how can we find its eigenvalues? To do this, we first calculate determinant of A minus lambda I, where I is an identity matrix that is the same size as A. Then we find the values of lambda that are solutions to the equation determinant of A minus lambda I equals 0. For an n x n matrix there will be n eigenvalues. However, not all of these eigenvalues have to be distinct or real. Let's work through an example. Consider the following matrix A. First we want to calculate the determinant of A minus lambda I. Using the equation for finding the determinant of a 2x2 matrix, we arrive at the following expression for the determinant. Note that this expression is a quadratic function in terms of the unknown eigenvalue lambda. Next we want to solve the equation determinant of A minus lambda I equals 0. In other words, we want to find the root of the quadratic equations we found earlier. The roots are lambda 1 equals 1, and lambda 2 equals 3. As expected, we found two eigenvalues for this 2x2 matrix. In this case, we were able to find two distinct real eigenvalues. We can now find eigenvectors that correspond to each eigenvalue. To do this, we want to solve the matrix Av equals lambda v, or A minus lambda I times v = 0. There were always be at least one eigenvector for each eigenvalue of the matrix. However, if some eigenvalues are repeated, there might be an infinite number of eigenvectors for that eigenvalue. Let's find the eigenvectors of the matrix we looked at earlier. First consider the eigenvalue 1. To find the corresponding eigenvector, we solve the expression A minus lambda 1 I times v1 = 0, where v1 is the unknown eigenvector we are looking for. To correspond to the eigenvalue, lambda 1 equals 1. We can simplify the term in parentheses on the left-hand side using the values of A and lambda 1. We can see that v1 = 1, -1 is a solution. To find the second eigenvector, we would simply repeat this process for lambda 2 equals 3.