﻿ The Computation of Functions of Square Matrices
San José State University

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Thayer Watkins
Silicon Valley
USA

The Computation of Functions
of Square Matrices

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

Proof:

Note that

#### (CXC−1)² = (CXC−1)(CXC−1) = (CXC−1CXC−1) hence (CXC−1)² = CX²C−1and furthermore (CXC−1)j = CXjC−1

Therefore term-by-term

#### Σj=0gj(CXC−1)j = Σj=0gj(CXjC−1) = C[Σj=0gjXj]C−1) which is the same as G(CXC−1) = CG(X)C−1

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

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

Proof:

Note that D² is equal to Diagonal {djj²} and hence Dj=Diagonal {djjj}. Therefore

## Theorem 3: If X is an n×n matrix with n distinct eigenvalues (λj, 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(λj)}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.)