Mathematical Methods for Physicists: A concise introduction - Site Map

ORTHOGONAL MATRIX (REAL)
Table 3.1.
Operations on matrices
Operation Matrix element ~A ~B If ~B ˆ ~A
Transposition ~B ˆ ~A T b ij ˆ a ji m n n m Symmetric a
Complex conjugation ~B ˆ ~A* b ij ˆ a* ij m n m n Real
Hermitian conjugation ~B ˆ ~A T * b ij ˆ a* ji m n n m Hermitian
a For square matrices only.
Evidently all the elements along the principal diagonal of a hermitian matrix must
be real.
A hermitian matrix is also de®ned as a matrix whose transpose equals its
complex conjugate:
~A T ˆ ~A* …that is; a kj ˆ a* jk †: …3:29a†
These two de®nitions are the same. First note that the elements in the principal
diagonal of a hermitian matrix are always real. Furthermore, any real symmetric
matrix is hermitian, so a real hermitian matrix is a symmetric matrix.
The product of two hermitian matrices is not generally hermitian unless they
commute. This is because of property (3.28): even if Ay ~ ˆ ~A and ~B y ˆ ~B,
… ~A ~B† y 6ˆ ~A ~B unless the matrices commute.
A matrix ~A that obeys
~A y ˆ A ~ …3:30†
is called an anti-hermitian (or skew-hermitian) matrix. All the elements along the
principal diagonal must be pure imaginary. An example is
0
1
6i 5 ‡ 2i 6 ‡ 3i
B
C
@ 5 ‡ 2i 8i 1 2i A:
6 ‡ 3i 1 2i 0
We summarize the three operations on matrices discussed above in Table 3.1.
Orthogonal matrix (real)
A matrix ~A ˆ…a jk † mn
satisfying the relations
~A ~A T ˆ ~I n ; …3:31a†
~A T ~A ˆ ~I m …3:31b†
is called an orthogonal matrix. It can be shown that if ~A is a ®nite matrix satisfying
both relations (3.31a) and (3.31b), then ~A must be square, and we have
~A ~A T ˆ ~A T ~A ˆ ~I: …3:32†
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