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2.2 A Finite Difference Approximation
55
Corollary 2.1 The system of equations defined by (2.14)–(2.15), has a
unique solution that can be computed using Algorithm 2.1.
At this point it should be noted that this result is valid only in the pres-
ence of exact arithmetics. On computers with a fixed number of digits rep-
resenting each real number, round-off errors may accumulate and destroy
the results of the algorithm. Precise results are available stating sufficient
conditions on the matrix in order for Gaussian elimination to provide a
good approximation to the solution of the linear system. Techniques also
exist to reduce the effect of round-off errors. These issues are discussed in
books on numerical linear algebra. If you are interested, you should con-
sult e.g. Golub and van Loan [11]. In the present course we regard these
difficulties, or more precisely, potential difficulties, as beyond our scope.
2.2.5 Positive Definite Matrices
Above, we showed that if the system is diagonal dominant, then Algorithm
2.1 is applicable. Now we will show that a similar result holds for positive
definite matrices.
Let us first briefly recall some basic facts concerning positive definite
matrices.
• A symmetric matrix A ∈ R n,n is referred to as positive definite if
T
n
v Av ≥ 0 for all v ∈ R ,
with equality only if v =0.
• A symmetric and positive definite matrix is nonsingular.
• A symmetric matrix is positive definite if and only if all the eigenval-
ues are real and strictly positive.
These and other properties of matrices are discussed in Project 1.2, and
can, of course, be found in any textbook on linear algebra. 6
The properties of symmetric and positive definite matrices are closely
connected to the similar properties for differential operators. These con-
nections will be studied below. In the present section we will prove that if
the matrix is symmetric and positive definite, the linear system of equations
can be solved by Algorithm 2.1.
Let us start by observing that a symmetric and positive definite matrix
is not necessarily diagonal dominant. Consider the 2 × 2 matrix

52
A = .
21

6 The basic concepts of linear algebra are introduced e.g. in the book of H. Anton [1]
In numerical linear algebra, the book of Golub and van Loan [11] is a standard reference.