Question

How can I think about matrix entries in a general sense?

I a looking for a much deeper analysis than “systems of equations.”

For instance, I know that when it is a rotation matrix, I know that the column vectors in the matrix, R, will be where the basis vectors land, and hence it will rotate any given vector accordingly. (is this correct in a general sense, as far as rotation matrices go?)

However, for a general linear transformation, I am struggling to find the deep significance of both the off-diagonal and diagonal entries. Would a good way to look at this be to think in terms of parallel and perpendicular components? For instance, the diagonal entries are those in which the basis vectors of the space are parallel to the corresponding column vector?

I am asking as a physics major. I am in my upper division E&M class, and I am studying susceptibility tensors, but I want to be able to visualize and deeply understand the purpose of each component in a matrix does, in a more general sense. Thanks!

Answer 1

Matrix representation is used to represent any system of interest. It contains the information about the system like the physical measurement (eigenvalues), eigenstates etc. Let us suppose that a number of students are sitting in a class, so each of them has a particular position (along a particular row and column); mathematically this can be represented in a matrix form. A matrix representation depends upon the choice of basis. The eigenstates of the matrix diagonalize itself giving eigenvalues along the main diagonal.