Minimizing r largest eigenvalues of A(x) can be solved by the program

minimize rt + Trace(X)

s.t. tI + X - A(x) >= 0

X >= 0

The variables are x(in R^k), t (in R) and X=X’ (in R^{pxp}).

This SDP can be expressed as LMI with 2p x 2p matrices. How do I express the constraints tI + X - A(x) >= 0 as a 2p x 2p matrix inequality.

Is the following constraint matrix correct

[ I X ]

[ X^T tI - A ]

I would appreciate if someone can give me a hint on the solution. Thanks