- Sylvester's law of inertia
linear algebra, Sylvester's law of inertia is a theoremdescribing a canonical representative for a real symmetric matrix under congruence transformations. It is named for J. J. Sylvesterwho stated and proved it in 1852.
The theorem states that a real symmetric matrix is congruent to exactly one diagonal matrix with diagonal entries all being +1,-1 or zero.
The "inertia" is defined as the triple containing the numbers of diagonal entries which are +1, -1 and 0 respectively. These numbers are equal to the numbers of positive, negative and zero
eigenvalues of "A": see also signature (quadratic form). A congruence transformation of "A" is formed as the product
where "S" is a
non-singular matrix. In other words, the signature of "A" as quadratic formis well-definedand independent under congruence transformations.
* [http://planetmath.org.sixxs.org/encyclopedia/SylvestersLaw.html Sylvester's law] on
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