San José State University 
appletmagic.com Thayer Watkins Silicon Valley & Tornado
Alley USA 
The Computation of Functions
of Square Matrices


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^{−1}CXC^{−1})
hence
(CXC^{−1})² = CX²C^{−1}
and furthermore
(CXC^{−1})^{j} = CX^{j}C^{−1}
Therefore termbyterm
Σ_{j=0}g_{j}(CXC^{−1})^{j} = Σ_{j=0}g_{j}(CX^{j}C^{−1})
= C[Σ_{j=0}g_{j}X^{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_{jj}²} and hence D^{j}=Diagonal {d_{jj}^{j}}.
Therefore
G(D) = Diagonal {Σ_{j=0}g_{j}d_{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.)