Matriz anti-hermitiana

Em álgebra linear, uma matriz quadrada com entradas complexas é dita anti-hermitiana ^{(português brasileiro)} ou anti-hermítica ^{(português europeu)} se sua conjugada transposta é a negativa da matriz original.^[1] Isto é, a matriz $A$ é anti-hermitiana se satisfaz a relação

$a_{ij}=-{\overline {a_{ji}}}$

para todo $i$ e $j$ , onde $a_{ij}$ é o elemento na linha $i$ e coluna $j$ de $A$ , e a barra superior denota o complexo conjugado.

Matrizes anti-hermitianas podem ser entendidas como versões complexas de matrizes antissimétricas, ou como análogo matricial de números puramente imaginários.^[2] O conjunto de todas as matrizes anti-hermitianas $n\times n$ forma a álgebra de Lie $u(n)$ , que corresponde ao grupo de Lie U(n). O conceito pode ser generalizado para incluir transformações lineares de qualquer espaço vetorial complexo com uma norma sesquilinear.

Notar que o adjunto de um operador depende do produto escalar considerado sobre o espaço real ou complexo $n$ dimensional $K^{n}$ . Se $(\cdot |\cdot )$ denota o produto escalar sobre $K^{n}$ , então afirmar que $A$ é anti-adjunta significa que para todo $u,v\in K^{n}$ tem-se $(Au|v)=-(u|Av)\,$ .

Exemplo

Por exemplo, a seguinte matriz é anti-hermitiana

A={\begin{bmatrix}-i&2+i\\-2+i&0\end{bmatrix}}

pois

-A={\begin{bmatrix}i&-2-i\\2-i&0\end{bmatrix}}

é o transposto conjugado de $A$ .

Referências

↑ Horn & Johnson (1985), §4.1.1; Meyer (2000), §3.2
↑ Horn & Johnson (1985), §4.1.2

Ver também

Bibliografia

Horn, Roger A.; Johnson, Charles R. (1985), Matrix Analysis, ISBN 978-0-521-38632-6, Cambridge University Press .
Meyer, Carl D. (2000), Matrix Analysis and Applied Linear Algebra, ISBN 978-0-89871-454-8, SIAM .

v d e Classes de matriz
Elementos explicitamente restritos	(0,1) Alternante Antidiagonal Anti-hermitiana Antissimétrica Flecha Banda Bidiagonal Binária Bissimétrica Bloco Booliana Cauchy Diagonal Elementar Hermitiana Hessenberg Lógica Markov Moore Não negativa Particionada Permutação Positive Anti-hermitiana Esparsa Simétrica Toeplitz Triangular Unitária Vandermonde
Constante	Troca Hilbert Identidade Lehmer Uns Pascal Pauli Redheffer Shift Zero
Condições sobre autovalores e autovetores	Companheira Convergente Defectiva Diagonalizável Hurwitz Positiva definida Estabilidade Stieltjes
Satisfazendo condições sobre produtos ou inversas	Congruente Idempotente ou Projeção Inversa Involutória Nilpotente Normal Ortogonal Ortonormal Singular Unimodular Unipotente Totalmente unimodular Peso
Com aplicações específicas	Adjunta Sinal alternante Aumentada Bézout Carleman Cartan Circulante Cofator Comutação Coxeter Derrogatória Distâncias Duplicação Eliminação Distância euclidiana Fundamental (equação diferencias linear) Geradora Gram Hessiana Householder Jacobiana Momento Pick Aleatória Rotação Seifert Cisalhamento Similaridade Simplética Totalmente positiva Transformação Wedderburn X–Y–Z
Usada em estatística	Bernoulli Centro Correlação Covariância Dispersão Duplamente estocástica Informação de Fisher Projeção Precisão Estocástica Transição
Usada em teoria dos grafos	Adjacência Biadjacência Grau Edmonds Incidência Laplaciana Adjacência de Seidel Tutte
Usada em ciência e engenharia	CKM Densidade Gama Gell-Mann Hamiltoniana Irregular S Transição de estado Substituição Z (química)
Termos relacionados	Forma canônica de Jordan Independência linear Exponencial matricial Wronskiano
Categoria:Matrizes

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