Diagonalization_Transformation

Matrix polynomials calculation using diagonalization transformation.

What is Diagonalization Transformation?

Diagonalization Transformation is the process of transforming
a matrix into diagonal form. Diagonal matrices represent
the eigenvalues of a matrix in a clear manner.
A Square matrix A of order n over a field F is said to be diagonalizable,
if it is similar to a diagonal matrix over the field F.
i.e
D = P^-1AP, or P^-1AP = diag(x₁,x₂....x_n)
where x₁,x₂....x_n are the eigen values of matrix A.

Conditions for Diagonalization :

1. Square Matrix A of dimension ‘n’ is diagonalizable if
A has ‘n’ linearly independent eigen vectors.
2. A matrix A is also diagonalizable if minimal
polynomial is the product of distinct linear factors.

Matrix Polynomials :

General form of matrix polynomail : f(A) = a_nAⁿ + a_n-1A^n-1 + .....+ kI

But, we don't calculate this polynomial by computing all the different powers of A.
Intsead of A, we compute different powers of diagonal matrix D
which is a very easy & time saving task in comparison to the former one.

To get the value of f(A) we use the following relation.
f(A) = Pf(D)P^-1
where P is the modal matrix, matrix of eigen vectors of A, and
D is the diagonal/spectral matrix, matrix of eigen values of A.