Metric signature (original) (raw)
Die Signatur (auch Trägheitsindex oder Index) ist ein Objekt aus der Mathematik, das vor allem in der linearen Algebra aber auch in unterschiedlichen Bereichen der Differentialgeometrie betrachtet wird. Genau handelt es sich um ein Zahlentripel, das eine Invariante einer symmetrischen Bilinearform ist. Dieses Zahlentripel ist also insbesondere unabhängig von der Basiswahl, bezüglich der die Bilinearform dargestellt wird. Grundlegend für die Definition der Signatur ist der Trägheitssatz von Sylvester, benannt nach dem Mathematiker James Joseph Sylvester. Daher wird die Signatur manchmal auch Sylvester-Signatur genannt.
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| dbo:abstract | Die Signatur (auch Trägheitsindex oder Index) ist ein Objekt aus der Mathematik, das vor allem in der linearen Algebra aber auch in unterschiedlichen Bereichen der Differentialgeometrie betrachtet wird. Genau handelt es sich um ein Zahlentripel, das eine Invariante einer symmetrischen Bilinearform ist. Dieses Zahlentripel ist also insbesondere unabhängig von der Basiswahl, bezüglich der die Bilinearform dargestellt wird. Grundlegend für die Definition der Signatur ist der Trägheitssatz von Sylvester, benannt nach dem Mathematiker James Joseph Sylvester. Daher wird die Signatur manchmal auch Sylvester-Signatur genannt. (de) In mathematics, the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r= 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively. The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature (v, 0). A Lorentzian metric is a metric with signature (p, 1), or (1, p). There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v − p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit. For example, s = 1 − 3 = −2 for (+, −, −, −) and its mirroring s' = −s = +2 for (−, +, +, +). (en) In matematica, e più precisamente in algebra lineare, la segnatura è una terna di numeri che corrispondono al numero di autovalori di una matrice simmetrica (o di un prodotto scalare associato). La segnatura è utile a determinare le proprietà essenziali di un prodotto scalare. Ad esempio, un prodotto scalare definito positivo, come quello presente in uno spazio euclideo, ha segnatura , mentre lo spazio-tempo di Minkowski (fondamentale nella teoria della relatività) ha segnatura oppure , a seconda delle convenzioni. (it) 数学、とくに線型代数学における符号数（ふごうすう、英: signature）は固有値の符号（正・負・零）を重複度を込めて数えたものである。 (ja) In natuurkunde, meer bepaald algemene relativiteitstheorie, bedoelt men met de signatuur van een metrische tensor het verschil in het aantal positieve en negatieve eigenwaarden van de metrische tensor. Indien er p positieve en q negatieve eigenwaarden zijn, is de signatuur , maar soms is men meer specifiek, en heeft het dan over een (p,q)-signatuur. Indien , spreekt men van een Euclidische signatuur. Indien er slechts één negatieve (of slechts één positieve) eigenwaarde is, kan men dit interpreteren als een unieke tijdscoördinaat, en spreekt men van een Lorentziaanse signatuur, zie ook minkowskitensor. Het begrip signatuur wordt ook wel index genoemd in lineaire algebra. (nl) 계량 부호수(計量符號數, 영어: metric signature)는 미분기하학에서 쓰이는 용어로, 계량 텐서의 양수 및 음수 고윳값들의 개수(중복도를 고려함)를 말한다. 보다 일반적으로 비퇴화 대칭 쌍선형 형식(이차 형식으로 볼 수 있음)에 대해 정의될 수 있다. 계량 부호수는 계량 텐서에 대응되는 실계수 대칭행렬을 한 뒤, 대각항들의 계수들 중에 양수인 것들과 음수인 것들의 개수를 센 것이다. 예를 들어 고윳값들이 -1, 2, 3, 6일 경우, 이를 양수가 셋, 음수가 하나라는 뜻에서 (3,1)로 쓰기도 하고, 더 구체적으로 (-,+,+,+)로 표기하기도 한다. 계량 부호수는 양수 고윳값의 개수 p와 음수 고윳값의 개수 q에 따라 다음과 같이 분류한다. * 만약 q=0이면, 부호수를 양의 정부호(陽-定符號, 영어: positive-definite)라고 한다. * 만약 p=0이면, 부호수를 음의 정부호(陰-定符號, 영어: negative-definite)라고 한다. * 둘 다 0이 아니면 이 부호수가 부정부호(不定符號, 영어: indefinite)라고 한다. 리만 계량은 양의 정부호 (p,0)인 계량이며, 은 부정부호 (p,1) 또는 (1,q) 부호수를 갖는 계량이다. 특이행렬의 경우 0을 고윳값으로 가질 수 있으므로 이를 포함해 로도 쓴다. (ko) A assinatura de um tensor métrico (ou mais geralmente um não degenerado forma simétrica bilinear, entendido como forma quadrática) é o número de valores próprios positivos e negativos da simétrica. Isto é, a matriz simétrica correspondente real é , e a entrada diagonal de cada sinal contado. Se a matriz é n vezes;n, o número possível de sinais positivos pode tomar qualquer valor p de 0 a n. A assinatura pode ser notada por qualquer par de inteiros tais como (P,e;Q), ou como uma lista explícita tal como (−,+,+,+) ou (+,−,−,−). A assinatura é dita ser "indefinida" ou "mista" se tanto "p" e "q" são maior ou menor mas não igual a zero(0). A é uma métrica com uma assinatura . Uma é uma com assinatura (p, - 1) (ou alguma vezes (1, - q)). Existe também outra definição de "assinatura" na qual usa um único número definido como o número "p - q", onde "p" e "q" são o número de valores próprios positivos e negativos do tensor métrico. Usando o tensor métrico não degenerado acima, a assinatura é simplesmente a soma de "p" e "- q". Por exemplo, para e para . (pt) В линейной алгебре сигнатура — числовая характеристика квадратичной формы или псевдоевклидова пространства, в котором скалярное произведение задано с помощью соответствующей квадратичной формы. (ru) Sygnaturą (p, q, r) tensora metrycznego nazywa się zespół liczb wskazujący, ile jest w tensorze metrycznym elementów dodatnich p, ujemnych q oraz zerowych r – jeżeli tensor ten jest sprowadzony do postaci diagonalnej. Sygnaturę nazywa się nieokreśloną lub mieszaną, jeżeli obie liczby p oraz q są niezerowe. Sygnaturę nazywa się zdegenerowaną, gdy r jest niezerowe. (pl) Сигнату́ра — числова характеристика квадратичної форми або псевдоевклідового простору, в якому скалярний добуток задано за допомогою відповідної квадратичної форми. (uk) |
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| rdfs:comment | Die Signatur (auch Trägheitsindex oder Index) ist ein Objekt aus der Mathematik, das vor allem in der linearen Algebra aber auch in unterschiedlichen Bereichen der Differentialgeometrie betrachtet wird. Genau handelt es sich um ein Zahlentripel, das eine Invariante einer symmetrischen Bilinearform ist. Dieses Zahlentripel ist also insbesondere unabhängig von der Basiswahl, bezüglich der die Bilinearform dargestellt wird. Grundlegend für die Definition der Signatur ist der Trägheitssatz von Sylvester, benannt nach dem Mathematiker James Joseph Sylvester. Daher wird die Signatur manchmal auch Sylvester-Signatur genannt. (de) In matematica, e più precisamente in algebra lineare, la segnatura è una terna di numeri che corrispondono al numero di autovalori di una matrice simmetrica (o di un prodotto scalare associato). La segnatura è utile a determinare le proprietà essenziali di un prodotto scalare. Ad esempio, un prodotto scalare definito positivo, come quello presente in uno spazio euclideo, ha segnatura , mentre lo spazio-tempo di Minkowski (fondamentale nella teoria della relatività) ha segnatura oppure , a seconda delle convenzioni. (it) 数学、とくに線型代数学における符号数（ふごうすう、英: signature）は固有値の符号（正・負・零）を重複度を込めて数えたものである。 (ja) В линейной алгебре сигнатура — числовая характеристика квадратичной формы или псевдоевклидова пространства, в котором скалярное произведение задано с помощью соответствующей квадратичной формы. (ru) Sygnaturą (p, q, r) tensora metrycznego nazywa się zespół liczb wskazujący, ile jest w tensorze metrycznym elementów dodatnich p, ujemnych q oraz zerowych r – jeżeli tensor ten jest sprowadzony do postaci diagonalnej. Sygnaturę nazywa się nieokreśloną lub mieszaną, jeżeli obie liczby p oraz q są niezerowe. Sygnaturę nazywa się zdegenerowaną, gdy r jest niezerowe. (pl) Сигнату́ра — числова характеристика квадратичної форми або псевдоевклідового простору, в якому скалярний добуток задано за допомогою відповідної квадратичної форми. (uk) In mathematics, the signature (v, p, r) of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. In relativistic physics, the v represents the time or virtual dimension, and the p for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r= 0, or as an explici (en) 계량 부호수(計量符號數, 영어: metric signature)는 미분기하학에서 쓰이는 용어로, 계량 텐서의 양수 및 음수 고윳값들의 개수(중복도를 고려함)를 말한다. 보다 일반적으로 비퇴화 대칭 쌍선형 형식(이차 형식으로 볼 수 있음)에 대해 정의될 수 있다. 계량 부호수는 계량 텐서에 대응되는 실계수 대칭행렬을 한 뒤, 대각항들의 계수들 중에 양수인 것들과 음수인 것들의 개수를 센 것이다. 예를 들어 고윳값들이 -1, 2, 3, 6일 경우, 이를 양수가 셋, 음수가 하나라는 뜻에서 (3,1)로 쓰기도 하고, 더 구체적으로 (-,+,+,+)로 표기하기도 한다. 계량 부호수는 양수 고윳값의 개수 p와 음수 고윳값의 개수 q에 따라 다음과 같이 분류한다. * 만약 q=0이면, 부호수를 양의 정부호(陽-定符號, 영어: positive-definite)라고 한다. * 만약 p=0이면, 부호수를 음의 정부호(陰-定符號, 영어: negative-definite)라고 한다. * 둘 다 0이 아니면 이 부호수가 부정부호(不定符號, 영어: indefinite)라고 한다. 특이행렬의 경우 0을 고윳값으로 가질 수 있으므로 이를 포함해 로도 쓴다. (ko) In natuurkunde, meer bepaald algemene relativiteitstheorie, bedoelt men met de signatuur van een metrische tensor het verschil in het aantal positieve en negatieve eigenwaarden van de metrische tensor. Indien er p positieve en q negatieve eigenwaarden zijn, is de signatuur , maar soms is men meer specifiek, en heeft het dan over een (p,q)-signatuur. Indien , spreekt men van een Euclidische signatuur. Indien er slechts één negatieve (of slechts één positieve) eigenwaarde is, kan men dit interpreteren als een unieke tijdscoördinaat, en spreekt men van een Lorentziaanse signatuur, zie ook minkowskitensor. (nl) A assinatura de um tensor métrico (ou mais geralmente um não degenerado forma simétrica bilinear, entendido como forma quadrática) é o número de valores próprios positivos e negativos da simétrica. Isto é, a matriz simétrica correspondente real é , e a entrada diagonal de cada sinal contado. Se a matriz é n vezes;n, o número possível de sinais positivos pode tomar qualquer valor p de 0 a n. A assinatura pode ser notada por qualquer par de inteiros tais como (P,e;Q), ou como uma lista explícita tal como (−,+,+,+) ou (+,−,−,−). (pt) |
| rdfs:label | Signatur (Lineare Algebra) (de) Segnatura (algebra lineare) (it) 계량 부호수 (ko) Metric signature (en) Signatuur (natuurkunde) (nl) 符号数 (ja) Sygnatura metryki (pl) Assinatura métrica (pt) Сигнатура (линейная алгебра) (ru) Сигнатура (лінійна алгебра) (uk) |
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