A matrix estimate with a semipositive imaginary part

This is the matrix argument (5.8) and the argument in Exercise 8 from the AN-04 lesson Positive Lagrangian ideals and distributions. All vectors in the estimate are real; transposes in the complex quadratic form are bilinear transposes.

The real symmetric diagonalization used for the nonnegative matrix is proved in Point sources and complex Gaussian kernels, Section 2. The finite algebra foundation, §10, and scalar foundation, §§12–13, provide matrix rules and compactness of the unit sphere.

Programme exposition by the AN-04 course contributors. CC0 1.0 Universal for the original expression.

The matrix estimate

For a complex symmetric invertible matrix , with , and a real , $-\operatorname{Im}\bigl((u-iY)^tA(u-iY)\bigr) =Y^tCY-u^tCu+2u^tBY. \tag{5.8}$ Choose , with small. The right side is at least . The last inequality holds because a zero vector for both quadratic terms satisfies , contradicting invertibility unless .

Quantitative bound. If the sum is zero, nonnegative symmetry gives , so , while directly. Thus , forcing . Compactness of the real unit sphere gives a positive lower bound for this sum. With , (5.8) equals . Choosing bounds it below by , which has a positive lower bound on that sphere. The same bound survives sufficiently small matrix perturbations.