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| ``Re: positive definite matrices''
by matte on 2003-08-26 15:30:10 |
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| In general, I don't believe there is any quick and easy way to
determine whether a matrix is positive definite. At least
without any prior knowledge of the matrix.
MathWorld gives some necessary conditions on the elements of a
matrix for the matrix to be positive definite. These are, however,
not sufficient. Therefore, using these one can - at most - prove that
a matrix is not positive definite. See
http://mathworld.wolfram.com/PositiveDefiniteMatrix.html
One of the conditions is that the determinant should be positive.
This is quite easy to understand: A matrix is positive definite
if and only if all it's eigenvalues are positive. Since the determinant
is the product of the eigenvalues the result follows.
Probably the easiest way to determine if a matrix is positive definite
is the calculate the determinants of all upper-left (or lower-right) sub
matrices. If all determinants are positive, (and only then) the matrix
is positive definite (again, see MathWorld). For example,
the two-two matrix
a b
c d
is positive definite if and only if ad-cb > 0 and a > 0.
Hope this helps.
Matte |
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