Note that there are several notations in common use for the complex conjugate. Applied physics and engineering texts tend to prefer , while most modern math and theoretical physics texts favor
. Unfortunately, the notation is also commonly used to denote adjoint
operators and the conjugate transpose
of a matrix. Because of these mutually contradictory conventions,
care is needed when consulting the literature. In this work, is used to denote the complex conjugate.
Common notational conventions for complex conjugate are summarized in the table below.
notation
references
This work; Abramowitz and Stegun (1972,
p. 16), Anton (2000, p. 528), Harris and Stocker (1998, p. 21), Golub
and Van Loan (1996, p. 15), Kaplan (1981, p. 28), Kaplan (1992, p. 572),
Krantz (1999, p. 2), Kreyszig (1988, p. 568), Roman (1987, p. 534),
Strang (1988, p. 220), Strang (1993)
Arfken (1985,
p. 356), Bekefi and Barrett (1987, p. 616), Press et al. (1989,
p. 397), Harris and Stocker (1998, p. 21), Hecht (1998, p. 18), Herkommer
(1999, p. 262)
In linear algebra, it is common to apply both the complex conjugate and transpose to the same matrix. The matrix obtained from a given matrix by this combined operation is commonly called the conjugate
transpose
of .
It is also called the Hermitian conjugate or Hermitian adjoint and, in some contexts,
the adjoint matrix. The adjugate
matrix is a different operation: it is the transpose
of the matrix of cofactors. In this work, denotes the conjugate
transpose matrix, while denotes an adjoint operator
when the context is operator theory.
is defined to be
is the matrix
obtained by replacing each element 
with its complex conjugate, 
(Arfken 1985, p. 210).
, while most modern math and theoretical physics texts favor
. Unfortunately, the notation 
is also commonly used to denote adjoint
operators and the conjugate transpose
of a matrix. Because of these mutually contradictory conventions,
care is needed when consulting the literature. In this work, 
is used to denote the complex conjugate.
by this combined operation is commonly called the conjugate
transpose 
of 
.
It is also called the Hermitian conjugate or Hermitian adjoint and, in some contexts,
the adjoint matrix. The adjugate
matrix is a different operation: it is the transpose
of the matrix of cofactors. In this work, 
denotes the conjugate
transpose matrix, while 
denotes an adjoint operator
when the context is operator theory.
