Hard Long Eigenvalued Days

I’ve had a few hard long days this week. It kind of annoys me when after Tuesday, I already feel tired. That’s before I’ve had any classes. Still, this week was an important week, with more assignments due tomorrow than any other week.

They are all finished now and I just noticed a few days ago that I have a holiday coming up next week for easter. The last three days were grueling. I woke up at 6:30 AM, my usual time but didn’t get back home well past 8PM. Today, I got home at 9PM. I leave the house at about 7:20AM. On Tuesday, I was locked in my office because my magnetic card didn’t work. It was actually a fault with the alarm system. I almost bashed my way out. The only thing that stopped me was a woman who was trying to get into the building. I was trying to get out. In the end, after 25 minutes, I located another exit and left. My office is like a maze. It’s the first floor of a condo building, dedicated to all sorts of small businesses.

I’ve chosen my subject for my study project for my Teaching College Mathematics class. It will involve a 10-15 page paper, including theory and an exercise, and a 20-25 minute oral presentation. I chose to do my project on eigenvalues and eigenvectors. Linear algebra has always been my favorite, even though Analysis is rapidly becoming a fave as well.

Given a linear transformation A, a non-zero vector x is defined to be an eigenvector of the transformation if it satisfies the eigenvalue equation Ax = λx for some scalar λ. In this situation, the scalar λ is called an eigenvalue of A corresponding to the eigenvector x.

Here is how we obtain eigenvectors:

The usual method of finding all eigenvectors and eigenvalues of a system is first to get rid of the unknown components of the eigenvectors, then find the eigenvalues, plug those back one by one in the eigenvalue equation in matrix form and solve that as a system of linear equations to find the components of the eigenvectors. From the identity transformation Ix = x, where I is the identity matrix, x in the eigenvalue equation can be replaced by Ix to give:

The identity matrix is needed to keep matrices, vectors, and scalars straight; the equation (A − λ) x = 0 is shorter, but mixed up since it does not differentiate between matrix, scalar, and vector.^[19] The expression in the right hand side is transferred to left hand side with a negative sign, leaving 0 on the right hand side:

The eigenvector x is pulled out behind parentheses:

This can be viewed as a linear system of equations in which the coefficient matrix is the expression in the parentheses, the matrix of the unknowns is x, and the right hand side matrix is zero. According to Cramer’s rule, this system of equations has non-trivial solutions (not all zeros, or not any number) if and only if its determinant vanishes, so the solutions of the equation are given by:

This equation is defined as the characteristic equation (less often, secular equation) of A, and the left-hand side is defined as the characteristic polynomial. The eigenvector x or its components are not present in the characteristic equation, so at this stage they are dispensed with, and the only unknowns that remain to be calculated are the eigenvalues (the components of matrix A are given, i. e, known beforehand).