By Patrick Suppes

Paperback with none writing within the booklet. released 1979 north-holland publishing corporation Amsterdam

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**Example text**

41) are called unitary matrices. 43) is a Hermitian matrix or self-adjoint matrix. 5 Eigenvalue Problems of Hermitian and Unitary Operators Hermitian and unitary matrices are most useful in physical applications. They have common important properties in the structure oftheir invariant}ubspaces. Let a general Hermitian or unitary matrix be represented by A, and the corresponding linear transformation operator by A. Then we have the following lemma. Lemma: If A transforms a subspace f/ of the vector space "f'" into itself, then it also transforms the complementary subspace (totally orthogonal subspace) f/l.

The product f(Jjf(Jj of two classes f(Jj and f(Jj is defined as the set that consists of the products of the elements of f(Jj and f(Jj. Note that the same element can appear several times in the product f(Jjf(Jj' In that event, it should be counted independently every time it appears. 25) ofthe group CJyo Their products can be calculated using the multiplication table: f(J2f(J2 = (C 3 + Ci1)(C 3 + Ci 1 ) = 2E + C 3 + Ci 1 = 2f(J1 + f(J2 f(J2f(J3 = (C 3 + C 3- 1 )(11 1 + 112 + (1 3 ) = 2(111 + 112 + (13) = 2f(J3 , .

5) in terms of its components x and y. Now we rotate the basis vectors i andj through an angle ex about the vertical axis. 1. 8) where R is the 2 x 2 orthogonal matrix -sinex ] . R- = R(ex) = [cosex . 7,8). Because i'V') is obtained from iU) by the rotation, we write i' = Ri V' = Rj), using the rotation operator R. 8) as [Ri, Rj] = [i,j]R . 10) Now consider how the vector r is transformed by this rotation R. As R is a linear operator, we have r' = Rr = R(xi + yj) = xRi + yRj = xi' + yj' = x(icosex + jsinex) + y( -isinex + jcosex) = (xcos ex - ysin ex)i + (x sin ex + ycos ex)j .