Sylvester's law of inertia
- Sylvester's law of inertia
In linear algebra, Sylvester's law of inertia is a theorem describing a canonical representative for a real symmetric matrix under congruence transformations. It is named for J. J. Sylvester who 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 form is well-defined and independent under congruence transformations.
ee also
*Metric signature
References
*
*
External links
* [http://planetmath.org.sixxs.org/encyclopedia/SylvestersLaw.html Sylvester's law] on PlanetMath.
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