Mathematical Methods for Physicists: A concise introduction - Site Map

MATRIX ALGEBRA
But if ~ A is an in®nite matrix, then ~ A is orthogonal if and only if both (3.31a) and
(3.31b) are simultaneously satis®ed.
Now taking the determinant of both sides of Eq. (3.32), we have (det ~A† 2 ˆ 1,
or det ~A ˆ1. This shows that ~A is non-singular, and so ~A 1 exists.
Premultiplying (3.32) by ~A 1 we have
~A 1 ˆ ~A T : …3:33†
This is often used as an alternative way of de®ning an orthogonal matrix.
The elements of an orthogonal matrix are not all independent. To ®nd the
conditions between them, let us ®rst equate the ijth element of both sides of
~A ~A T ˆ ~I; we ®nd that
X n
kˆ1
a ik a jk ˆ ij :
Similarly, equating the ijth element of both sides of ~A T ~A ˆ ~I, we obtain
X n
kˆ1
a ki a kj ˆ ij :
…3:34a†
…3:34b†
Note that either (3.34a) and (3.34b) gives 2n…n ‡ 1† relations. Thus, for a real
orthogonal matrix of order n, there are only n 2 n…n ‡ 1†=2 ˆ n…n 1†=2 di€erent
elements.
Unitary matrix
A matrix ~U ˆ…u jk † mn
satisfying the relations
~U ~U y ˆ ~I n ; …3:35a†
~U y ~U ˆ ~I m …3:35b†
is called a unitary matrix. If ~U is a ®nite matrix satisfying both (3.35a) and
(3.35b), then ~U must be a square matrix, and we have
~U ~U y ˆ ~U y ~U ˆ ~I: …3:36†
This is the complex generalization of the real orthogonal matrix. The elements of
a unitary matrix may be complex, for example
1
p 1 i
2 i 1
is unitary. From the de®nition (3.35), a real unitary matrix is orthogonal.
Taking the determinant of both sides of (3.36) and noting that
det ~U y ˆ…det ~U)*, we have
…det ~U†…det ~U†* ˆ 1 or jdet ~Uj ˆ1: …3:37†
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