- Sylvester's law of inertia
In

linear algebra ,**Sylvester's law of inertia**is atheorem describing a canonical representative for a real symmetric matrix undercongruence transformation s. It is named forJ. 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

eigenvalue s of "A": see alsosignature (quadratic form) . A congruence transformation of "A" is formed as the product:$SAS^T$

where "S" is a

non-singular matrix . In other words, the signature of "A" asquadratic form iswell-defined and independent under congruence transformations.**ee also***

Metric signature **References***

***External links*** [

*http://planetmath.org.sixxs.org/encyclopedia/SylvestersLaw.html Sylvester's law*] onPlanetMath .

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