Matrix Diagonalization & Eigen Decomposition

[NATURE.M]

Do these terms practically refer to the same thing?
Like a matrix is diagonalizable iff it can be expressed in the form A=PDP$^{-1}$, where A is n×n matrix, P is an invertible n×n matrix, and D is a diagonal matrix
Now, this relationship between the eigenvalues/eigenvectors is sometimes referred to as eigen decomposition? Can someone clarify these terms for me.

 

[PSarkar]

I think eigen decomposition is another term for spectral decomposition in the spectral theorem. Although it is stated in a different way than diagonalizing a matrix, the spectral decomposition is related to the Schur decomposition for normal linear transformations.

 

[AlephZero]

Now, this relationship between the eigenvalues/eigenvectors is sometimes referred to as eigen decomposition? Can someone clarify these terms for me.

For a matrix they are basically the same, but the concept of eigenvalues/vectors applies to other situations as well. For example the solution of ODEs and PDEs can involve infinite-dimensional vector spaces (and even uncountably infinite dimensional spaces), where "matrices" are not a very useful tool to work with.