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Sylvester's law of inertia
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<h2 id="statement">Statement </h2> <p>Let A {\displaystyle A} be a symmetric square matrix of order n {\displaystyle n} with <a href="/facts/Real_number/R02gw5Pb">real</a> entries. Any <a href="/facts/Non-singular_matrix/fPqXk3V8">non-singular matrix</a> S {\displaystyle S} of the same size is said to transform A {\displaystyle A} into another symmetric matrix B = S A S T {\displaystyle B=SAS^{\mathrm {T} }} , also of order n {\displaystyle n} , where S T {\displaystyle S^{\mathrm {T} }} is the transpose of S {\displaystyle S} . It is also said that matrices A {\displaystyle A} and B {\displaystyle B} are <a href="/facts/Matrix_congruence/L5nPV4BQ">congruent</a>. If A {\displaystyle A} is the coefficient matrix of some quadratic form of R n {\displaystyle \mathbb {R} ^{n}} , then B {\displaystyle B} is the matrix for the same form after the change of basis defined by S {\displaystyle S} . </p><p>A symmetric matrix A {\displaystyle A} can always be transformed in this way into a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> D {\displaystyle D} which has only entries 0 {\displaystyle 0} , + 1 {\displaystyle +1} , − 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 A {\displaystyle A} , i.e. it does not depend on the matrix S {\displaystyle S} used. </p><p>The number of + 1 {\displaystyle +1} s, denoted n + {\displaystyle n_{+}} , is called the positive index of inertia of A {\displaystyle A} , and the number of − 1 {\displaystyle -1} s, denoted n − {\displaystyle n_{-}} , is called the negative index of inertia. The number of 0 {\displaystyle 0} s, denoted n 0 {\displaystyle n_{0}} , is the dimension of the <a href="/facts/Kernel_(linear_algebra)/adhGUYa0">null space</a> of A {\displaystyle A} , known as the nullity of A {\displaystyle A} . These numbers satisfy an obvious relation </p> n 0 + n + + n − = n . {\displaystyle n_{0}+n_{+}+n_{-}=n.} <p>The difference, s g n ( A ) = n + − n − {\displaystyle \mathrm {sgn} (A)=n_{+}-n_{-}} , is usually called the signature of A {\displaystyle A} . (However, some authors use that term for the triple ( n 0 , n + , n − ) {\displaystyle (n_{0},n_{+},n_{-})} consisting of the nullity and the positive and negative indices of inertia of A {\displaystyle A} ; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data.) </p><p>If the matrix A {\displaystyle A} has the property that every principal upper left k × k {\displaystyle k\times k} <a href="/facts/Minor_(determinant)/CMudF5Eh">minor</a> Δ k {\displaystyle \Delta _{k}} is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence </p> Δ 0 = 1 , Δ 1 , … , Δ n = det A . {\displaystyle \Delta _{0}=1,\Delta _{1},\ldots ,\Delta _{n}=\det A.} <h2 id="statement-in-terms-of-eigenvalues">Statement in terms of eigenvalues</h2> <p>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 <a href="/facts/Matrix_congruence/L5nPV4BQ">congruent</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> ( B = S A S T {\displaystyle B=SAS^{\mathrm {T} }} , for some non-singular S {\displaystyle S} ). </p><p>The positive and negative indices of a symmetric matrix A {\displaystyle A} are also the number of positive and negative <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of A {\displaystyle A} . Any symmetric real matrix A {\displaystyle A} has an <a href="/facts/Eigendecomposition/a2nfF7hJ">eigendecomposition</a> of the form Q E Q T {\displaystyle QEQ^{\mathrm {T} }} where E {\displaystyle E} is a diagonal matrix containing the eigenvalues of A {\displaystyle A} , and Q {\displaystyle Q} is an <a href="/facts/Orthonormal/JfgL1nAh">orthonormal</a> square matrix containing the eigenvectors. The matrix E {\displaystyle E} can be written E = W D W T {\displaystyle E=WDW^{\mathrm {T} }} where D {\displaystyle D} is diagonal with entries 0 , + 1 , − 1 {\displaystyle 0,+1,-1} , and W {\displaystyle W} is diagonal with W i i = | E i i | {\displaystyle W_{ii}={\sqrt {\vert E_{ii}\vert }}} . The matrix S = Q W {\displaystyle S=QW} transforms D {\displaystyle D} to A {\displaystyle A} . </p> <h2 id="law-of-inertia-for-quadratic-forms">Law of inertia for quadratic forms</h2> <p>In the context of <a href="/facts/Quadratic_form/l2n1jXPe">quadratic forms</a>, a real quadratic form Q {\displaystyle Q} in n {\displaystyle n} variables (or on an n {\displaystyle n} -dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from x {\displaystyle x} to y {\displaystyle y} ) be brought to the diagonal form </p> Q ( x 1 , x 2 , … , x n ) = ∑ i = 1 n a i x i 2 {\displaystyle Q(x_{1},x_{2},\ldots ,x_{n})=\sum _{i=1}^{n}a_{i}x_{i}^{2}} <p>with each a i ∈ { 0 , 1 , − 1 } {\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 Q {\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 <a href="/facts/Definite_bilinear_form/utlBuVpN">positive definite</a> (respectively, negative definite) have the same <a href="/facts/Dimension_(linear_algebra)/z8m02A3W">dimension</a>. These dimensions are the positive and negative indices of inertia. </p> <h2 id="generalizations">Generalizations</h2> <p>Sylvester's law of inertia is also valid if A {\displaystyle A} and B {\displaystyle B} have complex entries. In this case, it is said that A {\displaystyle A} and B {\displaystyle B} are ∗ {\displaystyle *} -congruent if and only if there exists a non-singular complex matrix S {\displaystyle S} such that B = S A S ∗ {\displaystyle B=SAS^{*}} , where ∗ {\displaystyle *} denotes the <a href="/facts/Conjugate_transpose/7eBWtqVG">conjugate transpose</a>. In the complex scenario, a way to state Sylvester's law of inertia is that if A {\displaystyle A} and B {\displaystyle B} are <a href="/facts/Hermitian_matrix/qArPblYo">Hermitian matrices</a>, then A {\displaystyle A} and B {\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. </p><p><a href="/facts/Alexander_Ostrowski/r45sj25p">Ostrowski</a> proved a quantitative generalization of Sylvester's law of inertia:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> if A {\displaystyle A} and B {\displaystyle B} are ∗ {\displaystyle *} -congruent with B = S A S ∗ {\displaystyle B=SAS^{*}} , then their eigenvalues λ i {\displaystyle \lambda _{i}} are related by λ i ( B ) = θ i λ i ( A ) , i = 1 , … , n {\displaystyle \lambda _{i}(B)=\theta _{i}\lambda _{i}(A),\quad i=1,\ldots ,n} where θ i {\displaystyle \theta _{i}} are such that λ n ( S S ∗ ) ≤ θ i ≤ λ 1 ( S S ∗ ) {\displaystyle \lambda _{n}(SS^{*})\leq \theta _{i}\leq \lambda _{1}(SS^{*})} . </p><p>A theorem due to Ikramov generalizes the law of inertia to any <a href="/facts/Normal_matrices/WVKxrEK9">normal matrices</a> A {\displaystyle A} and B {\displaystyle B} :<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> If A {\displaystyle A} and B {\displaystyle B} are <a href="/facts/Normal_matrices/WVKxrEK9">normal matrices</a>, then A {\displaystyle A} and B {\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. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Metric_signature/w8d57bQG">Metric signature</a></li> <li><a href="/facts/Morse_theory/bJRqH8BQ">Morse theory</a></li> <li><a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky decomposition</a></li> <li><a href="/facts/Haynsworth_inertia_additivity_formula/x71a5V9k">Haynsworth inertia additivity formula</a></li></ul> <ul><li>Garling, D. J. H. (2011). <i>Clifford algebras. An introduction</i>. London Mathematical Society Student Texts. Vol. 78. Cambridge: <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-107-09638-7. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1235.15025">1235.15025</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://planetmath.org/SylvestersLaw">Sylvester's law</a> at <a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a>.</li> <li><a href="http://www.njohnston.ca/2009/08/sylvesters-law-of-inertia-and-congruence/">Sylvester's law of inertia and *-congruence</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Sylvester, James Joseph (1852). "A demonstration of the theorem that every homogeneous quadratic polynomial is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares" (PDF). Philosophical Magazine. 4th Series. 4 (23): 138–142. doi:10.1080/14786445208647087. Retrieved 2008-06-27. <a href="/wiki/James_Joseph_Sylvester" target="_blank">/wiki/James_Joseph_Sylvester</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 978-0-19-853248-4. <a href="978-0-19-853248-4" target="_blank">978-0-19-853248-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Carrell, James B. (2017). Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra. Springer. p. 313. ISBN 978-0-387-79428-0. <a href="978-0-387-79428-0" target="_blank">978-0-387-79428-0</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Ostrowski, Alexander M. (1959). "A quantitative formulation of Sylvester's law of inertia" (PDF). Proceedings of the National Academy of Sciences. A quantitative formulation of Sylvester's law of inertia (5): 740–744. Bibcode:1959PNAS...45..740O. doi:10.1073/pnas.45.5.740. PMC 222627. PMID 16590437. <a href="https://www.pnas.org/content/pnas/45/5/740.full.pdf" target="_blank">https://www.pnas.org/content/pnas/45/5/740.full.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Higham, Nicholas J.; Cheng, Sheung Hun (1998). "Modifying the inertia of matrices arising in optimization". Linear Algebra and Its Applications. 275–276: 261–279. doi:10.1016/S0024-3795(97)10015-5. <a href="https://doi.org/10.1016%2FS0024-3795%2897%2910015-5" target="_blank">https://doi.org/10.1016%2FS0024-3795%2897%2910015-5</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Ikramov, Kh. D. (2001). "On the inertia law for normal matrices". Doklady Mathematics. 64: 141–142. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>