TITLE: What's the point of orthogonal diagonalisation?
QUESTION [19 upvotes]: 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?

REPLY [5 votes]: Matrices are complicated objects.  At first glance they are rectangular arrays of numbers with a complicated multiplication rule.  Diagonalization helps us reduce the the matrix multiplication operation to a sequence of simple steps which make sense.  In your case you are asking about orthogonal diagonalization, so I will limit my comments to that.  Note that I normally think about diagonalization as a factorization of the matrix $\mathbf A$ as 
$$\mathbf A = \mathbf P \mathbf D \mathbf P^T$$
We know that $\mathbf D$ contains the eigenvalues of $\mathbf A$ and $\mathbf P$ contains the corresponding eigenvectors.  Consider the multiplication $\mathbf{y=Ax}$.  We can perform the multiplication $\mathbf y = \mathbf P \mathbf D \mathbf P^T \mathbf x$ step-by-step as follows:
First: $\mathbf x' = \mathbf P^T \mathbf x$.  This step projects $\mathbf x$ onto the eigenvectors because the eigenvectors are in the rows of $\mathbf P^T$.
Second: $\mathbf y' = \mathbf D \mathbf x'$.  This step stretches the resultant vector independently in each direction which corresponds to an eigenvector.  This is the key.  A matrix does a simple scaling (in general we may also shear or rotate) operation in independent directions (orthogonal case only!) in a particular basis or representation.
Third:  $\mathbf y = \mathbf P \mathbf y'$.  We take our resulting stretched vector and linearly combine the eigenvectors to get back to our original space.
So in summary, diagonalization tells us that under matrix multiplication, as vector can be represented in a special basis under which the actual operation of the matrix is a simple diagonal matrix (the simplest possible) and then represented back in our original space.