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2.5 Vector Spaces, Norms, and Inner Products 49

DEFINITION 2.5–9 All matrices in this definition are square and real.

T
• Positive Definite: A real n×n matrix A is positive definite if x Ax>0 for
all , x≠0.
• Positive Semidefinite: A real n×n matrix A is positive semidefinite if xT
Ax≥0 for all .
• Negative Definite: A real n×n matrix A is negative definite if x Ax<0 for
T
all , x≠0.
• Negative Semidefinite: A real n×n matrix A is negative semidefinite if x T
Ax≤0 for all .
T
• Indefinite: A is indefinite if x Ax>0 for some and x Ax<0 for
T
other .

Note that

where A s is the symmetric part of A. Therefore, the test for the definiteness
of a matrix may be done by considering only the symmetric part of A.

THEOREM 2.5–1: Let A=[a ij] be a symmetric n×n real matrix. As a result,
all eigenvalues of A are real. We then have the following

• Positive Definite: A real n×n matrix A is positive definite if all its
eigenvalues are positive.
• Positive semidefinite: A real n×n matrix A is positive definite if all its
eigenvalues are nonnegative.
• Negative Definite: A real n×n matrix A is negative definite if all its
eigenvalues are negative.
• Negative Semidefinite: A real n×n matrix A is negative semidefinite if all
its eigenvalues are non-positive.
• Indefinite: A real n×n matrix A is indefinite if some of its eigenvalues are
positive and some are negative.

Copyright © 2004 by Marcel Dekker, Inc.