Let x S1 + S2. Then x = q˜1 + q˜2 with q˜i Si. We write

x = q˜1 + Q1Q1q˜2 + (In − Q1Q1)q˜2 =: q1 + q2

with q1 = Q1a and q2 = Q2b = (In − Q1Q1)Q2b. Then

k(Q1Q1 − Q2Q2)xk2 = k(Q1Q1 − Q2Q2)(Q1a + Q2b)k2

=kQ1a + Q2Q2Q1a + Q2bk2

=k(In − Q2Q2)Q1a + Q2bk2

=a Q1(In − Q2Q2)Q1a

+2Re(a Q1(In − Q2Q2)Q2b) + b Q2Q2b

Thus, sin ϑ is the maximum of the Rayleigh quotient R(x) corresponding to Q1Q1 −Q2Q2, that is the largest eigenvalue of Q1Q1 −Q2Q2. As Q1Q1 −Q2Q2 is symmetric and positive semi-definite, its largest eigenvalue equals its norm,

Lemma 2.40 sin (S1, S2) = kQ2Q2 − Q1Q1k

Lemma 2.41 (S1, S2) = (S1 , S2 ).

Proof. Because

kQ2Q2 − Q1Q1k = k(I − Q2Q2) − (I − Q1Q1)k

the claim immediately follows from Lemma 2.40.

Bibliography

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[2]G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 2nd ed., 1989.

[3]R. ZURMUHL¨ , Matrizen und ihre technischen Anwendungen, Springer, Berlin, 4th ed., 1964.