Sylvester's law of inertia

Sylvester's law of inertia is a

. Namely, if ${\displaystyle A}$ is the symmetric matrix that defines the quadratic form, and ${\displaystyle S}$ is any invertible matrix such that ${\displaystyle D=SAS^{\mathrm {T} }}$ is diagonal, then the number of negative elements in the diagonal of ${\displaystyle D}$ is always the same, for all such ${\displaystyle S}$; and the same goes for the number of positive elements.

This property is named after James Joseph Sylvester who published its proof in 1852.^[1][2]

Statement

Let ${\displaystyle A}$ be a symmetric square matrix of order ${\displaystyle n}$ with

${\displaystyle S}$ of the same size is said to transform ${\displaystyle A}$ into another symmetric matrix ${\displaystyle B=SAS^{\mathrm {T} }}$, also of order ${\displaystyle n}$, where ${\displaystyle S^{\mathrm {T} }}$ is the transpose of ${\displaystyle S}$. It is also said that matrices ${\displaystyle A}$ and ${\displaystyle B}$ are congruent. If ${\displaystyle A}$ is the coefficient matrix of some quadratic form of ${\displaystyle \mathbb {R} ^{n}}$, then ${\displaystyle B}$ is the matrix for the same form after the change of basis defined by ${\displaystyle S}$.

A symmetric matrix ${\displaystyle A}$ can always be transformed in this way into a diagonal matrix ${\displaystyle D}$ which has only entries ${\displaystyle 0}$, ${\displaystyle +1}$, ${\displaystyle -1}$ along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of ${\displaystyle A}$, i.e. it does not depend on the matrix ${\displaystyle S}$ used.

The number of ${\displaystyle +1}$s, denoted ${\displaystyle n_{+}}$, is called the positive index of inertia of ${\displaystyle A}$, and the number of ${\displaystyle -1}$s, denoted ${\displaystyle n_{-}}$, is called the negative index of inertia. The number of ${\displaystyle 0}$s, denoted ${\displaystyle n_{0}}$, is the dimension of the null space of ${\displaystyle A}$, known as the nullity of ${\displaystyle A}$. These numbers satisfy an obvious relation

${\displaystyle n_{0}+n_{+}+n_{-}=n.}$

The difference, ${\displaystyle \mathrm {sgn} (A)=n_{+}-n_{-}}$, is usually called the signature of ${\displaystyle A}$. (However, some authors use that term for the triple ${\displaystyle (n_{0},n_{+},n_{-})}$ consisting of the nullity and the positive and negative indices of inertia of ${\displaystyle A}$; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data.)

If the matrix ${\displaystyle A}$ has the property that every principal upper left ${\displaystyle k\times k}$

${\displaystyle \Delta _{k}}$ is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence

${\displaystyle \Delta _{0}=1,\Delta _{1},\ldots ,\Delta _{n}=\det A.}$

Statement in terms of eigenvalues

The law can also be stated as follows: two symmetric square matrices of the same size have the same number of positive, negative and zero eigenvalues if and only if they are congruent^[3] (${\displaystyle B=SAS^{\mathrm {T} }}$, for some non-singular ${\displaystyle S}$).

The positive and negative indices of a symmetric matrix ${\displaystyle A}$ are also the number of positive and negative

of ${\displaystyle A}$. Any symmetric real matrix ${\displaystyle A}$ has an of the form ${\displaystyle QEQ^{\mathrm {T} }}$ where ${\displaystyle E}$ is a diagonal matrix containing the eigenvalues of ${\displaystyle A}$, and ${\displaystyle Q}$ is an square matrix containing the eigenvectors. The matrix ${\displaystyle E}$ can be written ${\displaystyle E=WDW^{\mathrm {T} }}$ where ${\displaystyle D}$ is diagonal with entries ${\displaystyle 0,+1,-1}$, and ${\displaystyle W}$ is diagonal with ${\displaystyle W_{ii}={\sqrt {\vert E_{ii}\vert }}}$. The matrix ${\displaystyle S=QW}$ transforms ${\displaystyle D}$ to ${\displaystyle A}$.

Law of inertia for quadratic forms

In the context of quadratic forms, a real quadratic form ${\displaystyle Q}$ in ${\displaystyle n}$ variables (or on an ${\displaystyle n}$-dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from ${\displaystyle x}$ to ${\displaystyle y}$) be brought to the diagonal form

${\displaystyle Q(x_{1},x_{2},\ldots ,x_{n})=\sum _{i=1}^{n}a_{i}x_{i}^{2}}$

with each ${\displaystyle a_{i}\in \{0,1,-1\}}$. Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of ${\displaystyle Q}$, i.e., does not depend on a particular choice of diagonalizing basis. Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is

. These dimensions are the positive and negative indices of inertia.

Generalizations

Sylvester's law of inertia is also valid if ${\displaystyle A}$ and ${\displaystyle B}$ have complex entries. In this case, it is said that ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle *}$-congruent if and only if there exists a non-singular complex matrix ${\displaystyle S}$ such that ${\displaystyle B=SAS^{*}}$, where ${\displaystyle *}$ denotes the conjugate transpose. In the complex scenario, a way to state Sylvester's law of inertia is that if ${\displaystyle A}$ and ${\displaystyle B}$ are Hermitian matrices, then ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle *}$-congruent if and only if they have the same inertia, the definition of which is still valid as the eigenvalues of Hermitian matrices are always real numbers.

Ostrowski proved a quantitative generalization of Sylvester's law of inertia:^[4][5] if ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle *}$-congruent with ${\displaystyle B=SAS^{*}}$, then their eigenvalues ${\displaystyle \lambda _{i}}$ are related by

where ${\displaystyle \theta _{i}}$ are such that ${\displaystyle \lambda _{n}(SS^{*})\leq \theta _{i}\leq \lambda _{1}(SS^{*})}$.

A theorem due to Ikramov generalizes the law of inertia to any

${\displaystyle A}$ and ${\displaystyle B}$:^[6] If ${\displaystyle A}$ and ${\displaystyle B}$ are , then ${\displaystyle A}$ and ${\displaystyle B}$ are congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.

See also

• Metric signature
• Morse theory
• Cholesky decomposition
• Haynsworth inertia additivity formula

References

• Garling, D. J. H. (2011). Clifford algebras. An introduction. London Mathematical Society Student Texts. Vol. 78. Cambridge:
  Zbl 1235.15025
  .

External links

• Sylvester's law at PlanetMath.
• Sylvester's law of inertia and *-congruence