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Lmin , the matrix R þ lQ has a single negative eigenvalue. We now use these facts to obtain a tighter lower bound on the value of the optimal Lagrange multiplier. 44) is negative. 44) as lÀ1 ¼ c 2j (À2 À lgj ) (1 þ lgj )2 À X c 2 (2 þ lg ) i i i=j (1 þ lgi )2 (1:45) where j denotes the index associated with this negative eigenvalue. 45) and solving lÀ1 ¼ c 2i (À2 À lgj ) (1 þ lgj )2 : This yields a quadratic equation in l l2 (c2j gj þ g2j ) þ 2l(gj þ c 2j ) þ 1 ¼ 0, (1:46) the roots of which are given by l¼ À1 + jcj j(gj þ c 2j )À1=2 : gj (1:47) By Lemma 2, the constraint cT xÃ !

Sin u1 . cos un À sin un cos u1 3 .. . 7 7 7 sin un 7 7: 7 7 7 5 cos un (1:75) 40 ROBUST MINIMUM VARIANCE BEAMFORMING The effect of premultiplying a direct sum-representation of a complex vector by T is to shift the phase of each component by the corresponding angle ui . 75) we have TxÀ1 TyÀ1 (F1 Tx x W F2 Ty y þ F3 Tx x W F4 Ty y) ¼ F1 x W F2 y þ F3 x W F4 y, (1:76) which does not hold for unitary matrices in general. We now compute rotation matrices Tb and Td such that the entries associated with Tb , we the imaginary components of products Tb b and Td d are zero.

Johnson and D. Dudgeon. Array Signal Processing: Concepts and Techniques. Signal Processing Series, Prentice Hall, Englewood Cliffs, 1993. 7. S. Haykin. Adaptive Filter Theory. Prentice Hall Information and System Sciences Series, Prentice Hall, Englewood Cliffs, 1996. 8. K. Harmanci, J. Tabrikian, and J. L. Krolik. Relationships between adaptive minimum variance beamforming and optimal source localization. IEEE Transactions on Signal Processing, 48(1), 1 – 13 (2000). 9. A. Ben-Tal and A. Nemirovski.

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