Difference between revisions of "Adjoint linear transformation"
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084016.png" /> is the matrix adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084018.png" /> is the [[Gram matrix|Gram matrix]] of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084019.png" />. | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084016.png" /> is the matrix adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084018.png" /> is the [[Gram matrix|Gram matrix]] of the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084019.png" />. | ||
| − | In a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084021.png" /> have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are | + | In a Euclidean space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010840/a01084021.png" /> have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are [[complex conjugate]]s. |
Revision as of 17:02, 30 November 2014
of a linear transformation 
The linear transformation 
on a Euclidean space (or unitary space) 
such that for all 
, the equality
 |
between the scalar products holds. This is a special case of the concept of an adjoint linear mapping. The transformation 
is defined uniquely by 
. If 
is finite-dimensional, then every 
has an adjoint 
, the matrix 
of which in a basis 
is related to the matrix 
of 
in the same basis as follows:
 |
where 
is the matrix adjoint to 
and 
is the Gram matrix of the basis 
.
In a Euclidean space, 
and 
have the same characteristic polynomial, determinant, trace, and eigen values. In a unitary space, their characteristic polynomials, determinants, traces, and eigen values are complex conjugates.
Comments
More generally, the phrase "adjoint transformation" or "adjoint linear mappingadjoint linear mapping" is also used to signify the dual linear mapping 
of a linear mapping 
. Here 
is the space of (continuous) linear functionals on 
and 
. The imbeddings 
, 
, 
connect the two notions. Cf. also Adjoint operator.
References
| [a1] | M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. Sect. 2 |
Adjoint linear transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Adjoint_linear_transformation&oldid=15342