For a linear algebra class, we are currently learning about finding the largest/smallest eigenvalues of a matrix using the power method and inverse iteration methods. I just want to make sure that I am understand how this works because the notes I have are a little confusing. Here are the notes I have found on the power method.
So you begin with an approximation vector, and this can be just about anything nonzero(?).
Then you keep multiplying the approximation matrix by the vector for however many iterations, each time factoring some scalar out from your resulting vector. For example in this book they seem to always use the y component of the vector to factor out from each iterations results when getting the approximation, but how are they choosing that? Are they just arbitrarily picking something? This is the part I don't understand the most.
Finally after the approximation seems to approach some limiting numbers, those are the numbers that make up the dominant eigenvector, and you can use that to find the corresponding eigenvalues
So as I understand it, this is exactly the same idea as power method except you subtract some number multiplied by the identity matrix from A, invert all of that, and that some number dictates that the approximated eigenvalue resulting from the algorithm will be the eigenvalue that is closest to whatever number you picked.
For example if I wanted the smallest eigenvalue, I would just set that number as 0, and the formula would be the exact same as power method except I multiply each iteration by the inverse of A rather than A. Am I understanding this correctly?
Sorry if my explanations or questions are bad; it's just that I had a hard time understanding it from class or reading the notes and I wanted some clarification.