Theorem 1: Let G( ) be a polynomial, finite or infinite, given by a sequence of coefficients {g_j, j=0 …}. Let X and C be n×n matrices. Then

G(CXC⁻¹) = CG(X)C⁻¹

Proof:

Note that

(CXC⁻¹)² = (CXC⁻¹)(CXC⁻¹) = (CXC⁻¹CXC⁻¹)
hence
(CXC⁻¹)² = CX²C⁻¹
and furthermore
(CXC⁻¹)^j = CX^jC⁻¹

Therefore term-by-term

Σ_j=0g_j(CXC⁻¹)^j = Σ_j=0g_j(CX^jC⁻¹)
= C[Σ_j=0g_jX^j]C⁻¹)
which is the same as
G(CXC⁻¹) = CG(X)C⁻¹

A diagonal matrix D is one such that d_jk=0 if j≠k. It is then given by the sequence {d_jj}.

Theorem 2: Let D be a diagonal matrix {d_jj and G a polynomial. Then G(D) is equal to Diagonal {G(d_jj)}.

Proof:

Note that D² is equal to Diagonal {d_jj²} and hence D^j=Diagonal {d_jj^j}. Therefore

G(D) = Diagonal {Σ_j=0g_jd_jj^j} = Diagonal {G(d_jj)};

Theorem 3: If X is an n×n matrix with n distinct eigenvalues (λ_j, j=1 to n) and C is the matrix of its eigenvectors then for a polynomial G

G(X) = G(CΛC⁻¹) = CG(Λ)C⁻¹) = C{G(λ_j)}C⁻¹)

Proof:

This follows from an application of Theorems 1 and 2.

All of the above, of couse, applies to the infinite polynomial Exp( ).

(To be continued.)