By Serre J.-P.

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**Extra info for Modular Forms of Weight 1 and Galois Repr., from Algebraic number fields**

**Example text**

Then det A < ( d e t a i l ) (det A22) • Since det A = det An det A/An and det An > (det A) / (det A22), there is a reversed Fischer inequality if A22 is nonsingular (for example, if A is positive definite): det {A/An) det{A/A22) < det A. As an application of the Fischer inequality, we give a determinantal inequality. Let A, 5 , C, D be square matrices of the same size, so that A C B \ f A* D J \ B* C ' * \ / AA* + 5 5 * AC* + BD'' \ D"" J ~ \ CA"" H- L>5* CC* -f DD* ) Then det( ^ ^"jl < d e t ( A A * + 5 5 * ) d e t ( C C * + L>J9*).

A-V2, Then C > 0, det C = (det B) / d e t A, and det{A + B) = d e t ( ^ i / 2 ( / + A-i/25A-i/2)ylV2^ = ( d e t ^ ) ( d e t ( / + C)) n = (det A) J]A, (/ + C) i=l n = (detA)[J(l + Ai(C)) > (detA)(l + t r C + detC) > (det A) (1 + det C) = det A + det B. The last inequality is an equality if and only if t r C = 0, that is, C = 0, since C is positive semidefinite, while C = 0 if and only if B = 0. 30) for a general A>0 follows by a continuity argument. 30) for Hadamard product [298, 341]. 17 (Oppenheim's Inequality) Let A = {aij) and B be nx n positive definite matrices.

M/E A computation shows that ) ^ ^^^ ~[ 0 A/E It follows that iLMRmXAU)^^^^ M % ) / ( ? On the other hand, Z{M/A)W ^ / ^ ) - W£;)/(A/E). = M/A^ so we again have the formula. SEC. 10) are the numbers of the positive, negative, and zero eigenvalues of A, respectively (including multiplicities). Of course, rank (A) — p{A) + q{A). By In(^) > (a, 6, c) we mean that p{A) > a, q{A) > 6, and z{A) > c. The inertia of a nonsingular Hermitian matrix and its inverse are the same since their (necessarily nonzero) eigenvalues are reciprocals of each other.

### Modular Forms of Weight 1 and Galois Repr., from Algebraic number fields by Serre J.-P.

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