What's the point of orthogonal diagonalisation? I've learned the process of orthogonal diagonalisation in an algebra course I'm taking...but I just realised I have no idea what the point of it is.
The definition is basically this: "A matrix $A$ is orthogonally diagonalisable if there exists a matrix $P$ which is orthogonal and $D = P^tAP$ where $D$ is diagonal". I don't understand the significance of this though...what is special/important about this relationship?

Mathematics is all about equivalence relations. Often we are not really interested in objects on-the-nose, we are interested in objects up to some natural equivalence relation. Matrices really define linear transformations, so in order to talk about linear transformations using matrices we need to specify when two matrices give the same abstract linear transformation. At the level of a first course in linear algebra, there are at least two natural choices, which are often not distinguished.


*

*Up to change of basis. This is a reasonable choice if your vector space has no natural norm or inner product (e.g. if it is a space of polynomials). The nicest matrices here are the ones that can be diagonalized (by an arbitrary invertible matrix). 

*Up to change of orthonormal basis. This is a reasonable choice if your vector space has an inner product / Euclidean norm, since you want to be able to compute inner products in the easiest possible way with respect to your basis. Here the nicest matrices are the ones that can be orthogonally diagonalized.


One of the most important examples of the latter is the use of Fourier series, which orthogonally diagonalize differentiation. This is an important technique for solving differential equations.

Perhaps you are also not familiar with the point of diagonalization. The point of diagonalization is to change coordinates so that the linear transformation you're interested in is as simple as possible (it doesn't get simpler than diagonal matrices). That makes it easy to analyze, as in the Fourier series example above. 
