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This is a good question. How much linear algebra do you have? Vectors are a mapping from the origin to a point in space. A matrix, when multiplied by that vector, transforms it - that is, the cross product of the matrix and the vector now map to a different point in space. Some matrices transform the vector by changing its magnitude but not its direction. These matrices are full matrices - but they have the same effect as just multiplying the vector by some ordinary scalar number. The vector with unchanged direction is called an "eigenvector" of the matrix and the amount the matrix scales it by is the "eignevalue" of the matrix. The term "eigen equation" is a little vague. Given a matrix, there are ways of finding its eigenvectros and eigenvalues. These are sometimes referred to as "eigen equations" for that matrix. There are also eigenfuctions. Eigenfunctions are best understood as the continuous case of the above discussion: it scales a function in such a way that it changes its magnitude but not its direction. As you have plainly realized by posting this question (one of the best in the biology group in a long time), eigenvalues, eigenvectors and eigenfunctions are very useful in computational science. Much of biophysics, molecular dynamics, and bioinformatics are precisely that: computational science. The sad answer is, the two paragraph explanations in biology textbooks tend not to be thorough or well presented. I can really only point you to the place where I learned it myself: For the basic intro to eigenvalues, try lecture 21 of this video series: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/. If you have no experience with linear algebra, the earlier videos are a good introduction. If you have had some experience with linear algebra, try the first nine lectures in this: http://ocw.mit.edu/courses/mathematics/18-085-computational-science-and-engineering-i-fall-2008/video-lectures/. The lecture videos take some time but they are worth it.
Also, if you have some experience with calculus rather than linear algebra, you might find the introduction to eigenvalues in Arthur Mattuck's differentials course more accessible: Lectures 24 on introduce them and their applications in terms of solving systems of ordinary differential equations: http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/.
That is why i wanted to ask the question in this group, but might a bit stupid of me not to say what i am suppose to use it for. But it is in the subfeld of molecular dynamics. I need to find some pKa for some aminoacids radikal in a enzyme. I made a experiment the other day, and the electrical charge of the enzyme changed differently then what I expected. Now i just need a method to finde these pKa either theoretical or practical.