# Sylvester's law of inertia

Sylvester's law of inertia(Sylvester's pronunciation,: Sylvester's law of inertia ) Describes some sort of nature that is immutable.

Specifically, for an arbitrary regular matrix S such that A and D = SAS 定義 defining a quadratic form, D , The number of positive components and the number of negative components aligned on the main diagonal line are the same regardless of S.

The name is due to proving this property in ().

## table of contents

Theorem argument

The n - th square matrix A is a symmetric matrix with real components. S of the same size shall convert A to another n-order symmetric matrix B = SAS.. here S ⊤ is B's. That is, the matrices A and B are mutually exclusive. If A is appropriate forR n _ B, then B will be in the same quadratic form S The coefficient matrix of the quadratic form obtained by performing the definition of.

In this way symmetric matrix A always has D such that the diagonal components are either 0, +1, -1 . Sylvester's law of inertia states that the number of such diagonal elements is the invariant of A (not dependent on how to take matrix S).

The number of +1 n _ + is calledpositive inertia index(_positive index of inertia ) and the number of -1 n _ - This is called the negative inertia index(_negative index of inertia). The number of 0 _ n _ 0 is the dimension of A, A (Degenerate order) of the number of floors. These are obvious

[{\ displaystyle n_ {0} + n _ {+} + n _ {-} n + n _ {+} + n _ {-} = n} = n}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/20f5ea4328bc370b92ff61e5798beb16501bc3af)

It has a relationship to be. The difference sign (A ) = _ n _ - - _ _ + is usually called (although the sign of the positive and negative inertia index and the triple of degenerate order (_ n _ 0, _n _ +, _ n _ -) There is also a document called a code number. For non - degenerate forms of a given degree, the same information is given in either case, but in general triplets have more information).

If the matrix A has the property that k _ x _ k _ main _ _ _ from the upper left are both nonzero, the negative inertia index is

Delta _ {0} = 1, \ Delta _ {1}, \ ldots, \ Delta _ {n} = \ det A}! [ {\ displaystyle \ Delta _ {0} = 1, \ Delta _ {1}, \ ldots, \ Delta _ {n} = \ det A}] (https://wikimedia.org/api / rest_v 1 / media / math / render / svg / a 437 e 4 b 1070 d 275 f 703869 d 5 ad 1 e 1 e f 6 aa 42 a 3 b 8)

Equal to the number of sign changes of

Paraphrasing argument using eigenvalues

The positive and negative inertia indices of the symmetric matrix A are also the positive and negative numbers of A. An arbitrary real symmetric matrix A uses a diagonal matrix E consisting of eigenvalues ​​of A and Q consisting of eigenvectors A _ = _ QEQ _ ⊤ has the form (). Furthermore, the matrix _ E _ = ( e _ ij ) is _ E _ = _ WDW , D is 0, +1, A diagonal matrix with -1 as a component, W can be a diagonal matrix with _w _ _ _ _ _ _ _ _ as components. Matrix _S _ = _ QW Converts D to A

Quadratic form of law of inertia

In the context of the real quadratic form Q of the real n-variable (or on the n-dimensional real vector space) is transformed into a diagonal form (a regular linear transformation)

{\ Displaystyle Q (x_ {1}, x_ {2}, \ ldots, x_ {n}) = \ sum _ {i = 1 (1, 2, …, xn) = Σ i = } {{} displaystyle Q \ (x_ {1}, x_ {2}, \ ldots, x_ {n} ) = \ \ {n} a_ {i} x_ { sum_ {i = 1} ^ {n} a_ {i} x_ {i} ^ {2}}] (https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe9aef2fd88df43d206c3ae6c7dbdafac262d5d)

. Let it be a _ i _ ∈ {0, 1, -1} respectively. Sylvester's relation law indicates that the number of codes given by this coefficient sequence is Q Argument that it is an invariant (independent of how to select diagonalized bases). Geometrically speaking, the law of inertia asserts that the dimension of an arbitrary maximum subspace such that the restriction of a given quadratic form is positive (or negative) is constant. The value of such dimension is equal to the positive (or negative) inertia index.

Generalization

Sylvester's law of inertia can also be described in the case where the matrix is ​​a complex component. In this case, let us assure that matrices A and B are * - congruent by means of an appropriate complex regular matrix S It is defined as _ B _ = _ SAS _ *. However, * represents.

Sylvester's law of inertia in the case of complex components is that the necessary and sufficient condition for A and B to be * - congruent is that their inertial indices are consistent.

This theorem was further generalized to that for pp. 141-142).

Theorem (Ikramov)

The necessary and sufficient condition for the congruence of the regular matrices A and B is that they have the same number of eigenvalues ​​on each open half line coming out of the origin of the Gaussian plane.

Related item

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References

1. ****Norman, C. W. (1986). _ Undergraduate algebra -.. Pp. 360 - 361.

• Sylvester, J J (1852). Philosophical Magazine (Ser. 4) 4(23): 138-142.:. Browsing June 27, 2008. .
• Ikramov, Kh. D. (2001). "On the inertia law for normal matrices". Doklady Math. 64: 141 - 142.
• Garling, D. J. H. (2011). Clifford algebras. An introduction. London Mathematical Society Student Texts. 78. Cambridge:

External link

• __- . (English) .. (English) . CS 1 maint: Multiple names: authors list

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Post Date : 2018-03-01 12:30

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