Diagonalization of linear transformations

Any matrix is a combination of rotation and scaling. A matrix is diagonalizable when it's just scaling, with no rotation component. A diagonal matrix is all scaling, along the coordinate axes. For example  the mtrix 
                                                       $\begin{pmatrix} 3 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 7 \end{pmatrix} $ 
scales any vector by a factor of $3$ along $x$ direction, by a factor of $2$ along $y$ direction, and by a factor of $7$ along $z$ direction.

But a diagonalizable matrix might scale directions other than $x,y,z$ directions, which is inconvenient. What we would like to do is make up a new coordinate system, where the basis vectors are the directions along which the matrix scales - ie, the eigenvectors. Then in this new coordinate system, the matrix will be diagonal, and its entries will be eigenvalues. So that's where D comes from. Suppose I want to multiply A times a vector x. Then to use my diagonalization trick, I need to (1) translate x into my new coordinate system, (2) scale it by the diagonal matrix D, and (3) translate it back into the original coordinate system. If P is the matrix where columns are eigenvectors, then to express x as a linear combination of eigenvectors, we need to
(Step I) solve
\[Py=x \Rightarrow y=P^{-1}x.\] 

In step 2, we scaley by $D$:
\[Dy=DP^{-1}x\]
In step 3, we turn it back to the original coordinate system:
\[P(DP^{-1}x)=PDP^{-1}x.\]

Observe that diagonalization has the same nature as doing a change of variable in $n$ dimensions where in the new coordinate systems the linear transformation become a diogonal matrix, and it naturally has numerous applications where it can help simplify computations or solve problems. For example see: 

1- https://users.math.msu.edu/users/kalfagia/h31.pdf

2- In quantum mechanics, any quantity which can be measured in a physical experiment, should be associated with a hermitian operator. For example, Hamiltonian is energy operator and it is represented by hermitian matrix. When you diagonalize hamiltonian in the main diagonal you will get energies of the system. And for practical purpose, hermitian and unitary matrices are diagonalizable by unitary matrices, see for example: 

math.purdue.edu/~eremenko/dvi/spectral.pdf

3- See also:  file:///Users/new-imachome/Downloads/Applicationsofdiagonalization.pdf

These are some applications of diagonalization. 

Note that any linear map $L: R^n \rightarrow R^n$ can be represented by a matrix $A_{n\times n}$ such that
\[L(x)=Ax.\]
So diagonalization of a linear operator is the same as diagonalization of a matrix.