In mechanics it is, for example, a way to find principal axes of inertia (with tensor of inertia being the diagonalized matrix).

One other thing that readily springs to mind is finding normal modes of an oscillating system (which requires simultaneous diagonalization of two matrixes of kinetic and potential energy)

All eigenvalues of the given matrix appeared in the diagonal of the diagonal matrix.

This is a method to get the eigen values for an algebric system of equations.

If P⁻¹AP = D, where D is a diagonal matrix then Am = PDmP⁻¹ for all integers m > 1.

For A = PDP⁻¹, if A is invertible then the inverse of A is 

A⁻¹ = (PDP⁻¹)⁻¹ = PD⁻¹P⁻¹.

If we have the diagonal form, the powers of the matrix A can be deduced easily. Consequently, the analytic functions f at that matrix (such as exp(A)) can be deduced. This remark is useful in writing the solution of linear differential system with constant coefficients, in terms of the matrix exp(tA) (or exp(zA)).

There are quite a lot of diagonalization applications.

See the text in the attachment.

I hope it will be useful for you.

Greetings, Tadeusz

Solving eigenequations is nothing else than diagonalization.  Eigenvalues found this way are natural frequencies of vibrations (of mechanical constructions) or energy levels (in quantum mechanics).

the main application is to uncouple equations (algebraic or differential) so that each equation contains only one variable. Therefore, it can be solved singly.

Such is mentionad above, ther are a lot of applications because the eigenvalues of an operator (finite or infinite dimensional) provides great information about the problem associated to the corresponding operator. For example, when one discretize a partial diferential equation, the stady of the satability is related with thet wo-norm of the amplification matrix and then, the stability of the method dpends on these eigenvalues.

Another application is to get the exponential of a matrix, which is applicable for the deduction of various geometric integrators.

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There is a lot of applications as mentionned above.

We can cite the dermination of natural frequency of vibrations (mechanics).

In quantum mechanics, we can have by this method the corresponding enrgy levels for a system

Diagonalization of a matrix is used to similarity.

Diagonalization of a matrix is used to similarity.

If many applications of diagonalization of a matrix have been mentioned, the reduction of quadratic forms is also one important application (you could find some examples in the chapter 6 of the following reference

http://www.unitheque.com/Livre/ellipses/References_sciences/Mathematiques-66174.html

unfortunately in french)

P⁻¹AP = D is the formula for diagonalisation which leads to another formula, A = PDP⁻¹, If you need to transform a curve or a surface, x, using the transformation, y = Ax, and you want the new curve or surface to be extended or rotated by some amount, you can get the required transformation matrix A by the formula A = PDP⁻¹. The matrices P, A = P⁻¹ and D can be chosen with respect to the requirement.

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

https://www.math.purdue.edu/~eremenko/dvi/spectral.pdf

The main purpose of diagonalization is determination of functions of a matrix.

If P⁻¹AP = D, where D is a diagonal matrix, then it is known that the entries of D are the eigen values of matrix A and P is the matrix of eigen vectors of A.

If the matrix A is invertible then A = PDP⁻¹

Then f(A) = P*f(D)*P⁻¹

This expression is very useful because f(D) is simply a diagonal matrix whose diagonal elements are f(D1), f(D2)........f(Dn) where D1, D2.......Dn are the eigen values of matrix A.

f can be any function such as sin, cosh, sqrt ...... etc. In essence diagonalization provides a simple pathway for evaluation of functions of a matrix.

finding the powers of A and also reducing quadratic forms to canonical forms by orthogonal transformations.