# Preface

A square n×n matrix A is called diagonalizable if it has n linearly independent eigenvectors. For such matrices, there exists a nonsingular (meaning its determinant is not zero) matrix S such that $${\bf S}^{-1} {\bf A} \,{\bf S} = {\bf \Lambda} ,$$ the diagonal matrix. Then we can define a function of diagonalizable matrix A as $$f({\bf A}) = {\bf S}\, f({\bf \Lambda}) \, {\bf S}^{-1} .$$

Introduction to Linear Algebra with Mathematica

# Monte Carlo Method

Simulation of the motion of a random particle may be used to approximate the solution to linear parabolic equation. In particular, consider the two dimensional initial boundary value problem:

$\begin{split} u_t &= L[u] + f(x,y,t), \qquad x,y \in R, \quad t> 0, \\ u &= \phi (x,y,t) , \qquad x,y \in \partial R, \quad t> 0, \\ u(t=0) &= g(x,y) , \qquad x,y \in R, \end{split}$
with the differential operator L[ ] defined by
$L[u] = A\,u_{xx} + 2B\,u_{xy} + C\,u_{yy} + D\,u_x + E\,u_y ,$
where A, B, C, D, E, F are all functions of {x,y,t}. Here R is two dimensional domain on ℝ² with smooth boundary ∂R.