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By Ali H. Sayed

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In terms of these Schur complements, it is easy to verify by direct calculation that the matrix M can be factored in either of the following two useful forms: Inertia. e,, M = M * ) , its eigenvalues are real (cf. Sec. l). We then define its inertia as the triplet In{M} = {I+, I - , l o } , where I+ ( M ) = the number of positive eigenvalues of M I- ( M ) = the number of negative eigenvalues of M I o ( M ) = the number of zero eigenvalues of M Congruence. An important result regarding congruent matrices is the following, which states that congruence preserves inertia.

Kaveh (University of Minnesota), L. Ljung (Linkoping University, Sweden), G. Maalouli (General Dynamics Corporation), S. K. Mitra (University of California at Santa Barbara), B. Ottersten (Royal Institute of Technology, Sweden), A. Petropulu (Drexel University), H. Sakai (Kyoto University, Japan), T. Soderstrom (Uppsala University, Sweden), P. Stoica (Uppsala University, Sweden), P. P. Vaidyanathan (California Institute of Technology), M. Vetterli (EPFL, Switzerland), and B. Wahlberg (Royal Institute of Technology, Sweden).

Then V,g = 0 and V,*g = /3. 3. Let g ( z ) = z * z , where z is acolumn vector. Then V,g = z* and Vz*g= z . + + + 4. Let g(z) = X a f t z * P z * r z , where X is a scalar, { a ,p } are column vectors and r is a matrix. Then V,g = a* z ' r and V,*g = p r z . g]. Consider again the example g(z) = X then its Hessian matrix is 0: g = r. , . , zn} and a symmetric matrix r, then the gradient of g with respect to z is again defined as + + so that we now obtain V,g = aT PT 2zTr. Likewise, the Hessian matrix becomes 02 g = 2r.

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