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Encyclopedia > Sylvester's law of inertia

In linear algebra, Sylvester's law of inertia states that the inertia of a symmetric matrix A is invariant under congruence transformations. It is named for J. J. Sylvester. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, an invariant is something that does not change under a set of transformations. ... In mathematics and especially in abstract algebra, a congruence relation or simply congruence is an equivalence relation that is compatible with some algebraic operation(s). ... James Joseph Sylvester James Joseph Sylvester (September 3, 1814 - March 15, 1897) was an English mathematician and lawyer. ...


The inertia of a symmetric matrix A is defined as the triple containing 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 In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, signature can refer to The signature of a permutation is ±1 according to whether it is an even/odd permutation. ...

SAST

where S is any given non-singular matrix. In other words, the signature of A as quadratic form is well-defined and independent of change of basis. In mathematics and especially linear algebra, an n-by-n matrix A is called invertible, non-singular or regular if there exists another n-by-n matrix B such that AB = BA = In, where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ... In linear algebra, we may consider some finite-dimensional vector space, which can have associated with it some basis with which we can work with respect to. ...


  Results from FactBites:
 
James Joseph Sylvester Summary (1857 words)
Sylvester was born on September 3, 1814 in London, the son of Abraham Joseph.
Sylvester had resigned from Woolwich in 1870 and was busily engaged in his mathematical work, but he could not resist an invitation from the eminent American physicist Joseph Henry to come to the newlyfounded Johns Hopkins University in Baltimore.
Sylvester died on March 15, 1897 and was buried at the Jewish Cemetery at Ball's Pond, London.
James Joseph Sylvester at AllExperts (868 words)
One of Sylvester's lifelong passions was for poetry; he read and translated works from the original French, German, Italian, Latin and Greek, and many of his mathematical papers contain illustrative quotes from classical poetry.
In 1877 Sylvester again crossed the Atlantic Ocean to become the inaugural professor of mathematics at the new Johns Hopkins University in Baltimore, Maryland.
Sylvester House, an undergraduate dormitory at Johns Hopkins, is named in his honor.
  More results at FactBites »


 

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