Suppose I have positive semi-definite matrices A and B. Then $[A$ $X;X$ $B]$$>$$=0$ for $X$=$A$^$0.5$$C$$B$^$0.5$, where $C$ is the contraction matrix with maximum eigenvalue less than 1.
Horn, Roger A.; Johnson, Charles R., Topics in matrix analysis, Cambridge etc.: Cambridge University Press. viii, 607 p. \sterling 45.00; \$ 59.50 (1991). ZBL0729.15001.
I have some questions:
(1) Is it possible for $C$ to have negative eigenvalues?
(2) Are their any properties of $C$ other than eigenvalue<1? (Please, suggest a book or something).
(3) Is it possible to compute the maximum bounds of the matrix $C$ (Where any random contraction is inside that maximum bound)?
I shall be very thankful for any guidance and suggestion. Thanks