Matakuliah
Tahun
: Matrix Algebra for Statistics
: 2009
POSITIVE DEFINITE AND NON-NEGATIVE
DEFINITE MATRICES
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Introduction
• Quadratic forms that are non-negative definite play an important role in statistical theory, particularly those related to chisquare distributions
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Definition
Let A be an n x n Hermitian matrix, and let x € C n . Then x*Ax is said to be a Hermitian n on -n egative d efinite
(n.n.d.) quadratic form if x*Ax ≥ 0 for all x.
If x*Ax is Hermitian n.n.d. we say that
A is Hermitian n.n.d. and written A ≈ 0
(semi-definite)
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If A is Hermitian and n.n.d., and there exists x, x ≠ 0 such that x*Ax = 0,
We say that A is Hermitian positive semidefinite or positive indefinite.
An alternative definition is that A is n.n.d. and det A = 0.
If x*Ax > 0 for all x ≠0, then we say that A is
Hermitian positive definite (p.d.) definite and write A > 0.
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Example
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