552 Chapter 19The matrix eigenvalue problem
The Hermitian conjugate matrix A
†The Hermitian conjugate of Ais the transpose of the complex conjugate,
A
†1 = 1 (A*)
T1 = 1 (A
T)* (19.53)
Thus, for order 3,
The Hermitian conjugate is also called the conjugate transpose matrix, the associate
matrix, and (in quantum mechanics) the adjoint matrix (not to be confused with the
matrix of the same name discussed in Section 18.4).
EXAMPLE 19.16Find the Hermitian conjugate of the matrix
The complex conjugateA*and Hermitian conjugateA
†are
0 Exercises 36–38
The Hermitian conjugate plays the same role for complex matrices as does the
transpose for real matrices. For example, the inner (scalar) product of two vectors
a 1 = 1 (a
1,a
2,a
3)andb 1 = 1 (b
1,b
2,b
3) in a complex vector space is defined as (compare
equation (18.62))
ab (19.54)
·==
ab
†()aaa
b
b
b
123123*** ==++ab ab ab
11 22 33AA
*
=−−−
+−
,=
−+
−
3231
12 2 2
312
23
ii
ii
ii
i
†22
−− 12
i
A=
++−
−
3231
12 2 2
ii
ii
if A=
aaa
aaa
aaa
11
12 1321 22 2331 32 33then A =
†aaa
aaa
aa
11 21 3112 22 3213 223 33**a