Matrix polynomials calculation using 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....xn)
where x1,x2....xn are the eigen values of matrix A.
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.
General form of matrix polynomail : f(A) = anAn + an-1An-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.