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.

<cite authors="Horn, Roger A.; Johnson, Charles R.">_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](https://zbmath.org/?q=an:0729.15001).</cite>

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