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The adjoint of a matrix (Read 5438 times)
Gerrit-Jan Linker
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The adjoint of a matrix
13.01.08 at 15:30:40
 
The adjoint of a matrix
 
The adjoint of a matrix is the transpose and conplex-conjugate of a matrix. Synonyms: conjugate transpose, Hermitian transpose, adjoint matrix  
 
Given a matrix A, the matrix AT is the transpose of A if A(i,j) = AT(j,i).
 
The complex-conjugate of a complex number z=a+bi is z=a-bi (where a and b are real numbers).
 
The complex-conjugate of a matrix is a matrix for which all elements A(i,j) are replaced by their complex-conjugates (A(i,j))*
 
The adjoint of a matrix A is denoted as A
 
For a product of two matrices A and B it is found that:
(AB) = BA
 
See also:
Conjugate transpose
http://en.wikipedia.org/wiki/Conjugate_transpose
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« Last Edit: 04.02.08 at 14:55:09 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
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Posts: 75
Re: The adjoint of a matrix
Reply #1 - 02.02.08 at 11:50:04
 
It is interesting when this property is applied to the eigenvalue equation: |α> = ωα|α>
 
|α> = ωα|α> <=>
(|α>) = (ωα|α>) <=>
|α> = |α>ω*α <=>
<α| = <α|ω*α
 
See also:
Dirac notation. Bra-ket notation
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1200256881
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« Last Edit: 04.02.08 at 14:56:22 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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