Eigenvectors are special vectors that, when multiplied by a matrix A, result in a scalar multiple of themselves. This is represented by the equation A times vector v equals lambda times vector v, where lambda is the eigenvalue. In this example, we have a 2 by 2 matrix A and two vectors. Notice how the blue vector, when transformed by matrix A, results in a scaled version of itself in the same direction. This makes it an eigenvector. The green vector, however, changes direction when transformed, so it is not an eigenvector.
To find eigenvectors, we first need to find the eigenvalues. We do this by solving the characteristic equation: determinant of A minus lambda times I equals zero, where I is the identity matrix. Let's work through our example with matrix A. We substitute the matrix and expand the determinant. After simplifying, we get the equation: 3 minus lambda squared minus 1 equals 0. Solving this equation gives us two eigenvalues: lambda equals 2 or lambda equals 4. These eigenvalues will be used in the next step to find the corresponding eigenvectors.
Now that we have our eigenvalues, we can find the eigenvectors. For each eigenvalue lambda, we solve the system (A minus lambda I) times vector v equals zero. Let's start with lambda equals 2. We substitute this value into our matrix equation. After subtracting 2 times the identity matrix from A, we get a new matrix. This gives us the equation v1 plus v2 equals 0, which means v1 equals negative v2. So the eigenvector for lambda equals 2 is any non-zero multiple of the vector negative 1, 1. Similarly, for lambda equals 4, we get the equation negative v1 plus v2 equals 0, which means v1 equals v2. So the eigenvector for lambda equals 4 is any non-zero multiple of the vector 1, 1. Notice that these two eigenvectors are perpendicular to each other, which is a special property of symmetric matrices.
Let's visualize what eigenvectors mean geometrically. An eigenvector is a vector that, when multiplied by a matrix, maintains its direction and is only scaled by its eigenvalue. For our example matrix, we found two eigenvectors: vector v1 equals 1,1 with eigenvalue 4, and vector v2 equals negative 1,1 with eigenvalue 2. Looking at the blue vector v1, when we multiply it by matrix A, we get the red vector which is exactly 4 times the original vector - pointing in the same direction but scaled by the eigenvalue 4. Similarly, the green vector v2, when multiplied by A, gives us the orange vector which is exactly 2 times the original - again, same direction but scaled by eigenvalue 2. This is the defining property of eigenvectors: they maintain their direction under the transformation, only changing in length.