PlanetMath: canonical basis for symmetric bilinear forms

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canonical basis for symmetric bilinear forms

(Definition)

If $B:V \times V \rightarrow K$ is a symmetric bilinear form over a finite-dimensional vector space, where the characteristic of the field is not 2, then we may prove that there is an orthogonal basis such that is represented by

$\displaystyle \bordermatrix{& \cr & a_{1} & 0 & \ldots & 0\cr & 0 & a_{2} & \ldots & 0\cr & \vdots & \vdots & \ddots & \vdots\cr & 0 & 0 &\ldots & a_{n}\cr }$

Recall that a bilinear form has a well-defined rank, and denote this by .

If $K = \mathbb{R}$ we may choose a basis such that $a_1 = \cdots = a_t = 1$ , $a_{t+1} = \cdots = a_{t+p} = -1$ and $a_{t+p+j} = 0$ , for some integers and , where $1 \le j \le n-t-p$ . Furthermore, these integers are invariants of the bilinear form. This is known as Sylvester's Law of Inertia. is positive definite if and only if , . Such a form constitutes a real inner product space.

If $K = \mathbb{C}$ we may go further and choose a basis such that $a_1 = \cdots = a_r = 1$ and $a_{r + j} = 0$ , where $1 \le j \le n-r$ .

If we may choose a basis such that $a_1 = \cdots = a_{r-1} = 1$ ,

or ; and $a_{r+j} = 0$ , where $1 \le j \le n-r$ , and is the least positive quadratic non-residue.

"canonical basis for symmetric bilinear forms" is owned by Mathprof. [ full author list (2) | owner history (1) ]

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Also defines:

Sylvester's Law of Inertia

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Cross-references: quadratic non-residue, positive, inner product space, real, positive definite, invariants, integers, rank, well-defined, bilinear form, basis, orthogonal, field, characteristic, vector space, finite-dimensional, symmetric bilinear form
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This is version 4 of canonical basis for symmetric bilinear forms, born on 2005-01-08, modified 2007-05-26.
Object id is 6631, canonical name is CanonicalBasisForSymmetricBilinearForms.
Accessed 2744 times total.

Classification:

AMS MSC:	15A63 (Linear and multilinear algebra; matrix theory :: Quadratic and bilinear forms, inner products)
	11E39 (Number theory :: Forms and linear algebraic groups :: Bilinear and Hermitian forms)
	47A07 (Operator theory :: General theory of linear operators :: Forms )

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