# Help! How to express log_det(I-w*w') >=A in CVX, where w is a matrix variable

How to express log_det(I-w*w’)>=A

use the determinant identity version of the Schur Complement:
det([I w;w' I]) = det(I)*det(I-w*inv(I)*w') .

So

variable w(n,m)
log_det([eye(n)  w;w' eye(m)]) >= A


should work. It will enforce [eye(n) w;w' eye(m)] being positive semidefinite, which is equivalent to I-w*w' being positive semidefinite. Without that, the constraint would not be convex.

Thank you for your kind help.

The original problem is
log_det(I+2*re(H'*W1*Y)-Y'*(I+H*W_2*W_2'*H'+H*W_3*W_3'*H'+H*W_4*W_4'*H')*Y)>=A,where’‘re’’ denotes the real part, H and Y are constant complex matrix, respectively. Here, W1,W2,W3, and W4 are the complex matrix variables that needs to be optimized. However, I don’t know how to generalize to this problem using the above method. Could you give me an idea when you are available? Thank you.

That looks like a big mess. Have you proven the constraint is convex?

f_{n,m_k}=\log\det\Big(\mathbf{I}+2\mathrm{Re}\left(\mathbf{H}^H_n\mathbf{V}_{m_k}\mathbf{Y}_{n,m_k}\right)-\mathbf{Y}^H_{n,m_k}\mathbf{J}_{n,m_k}\mathbf{Y}_{n,m_k}\Big) where \mathbf{J}_{n,m_k}=\mathbf{I}+\underset{\left(m',{k'}\right)\notin \mathcal{V}}{\sum_{m'=1}^{M}\sum_{k'=1}^{2}}\mathbf{H}^H_n\mathbf{V}_{m'_{k'}}\mathbf{V}^H_{m'_{k'}}\mathbf{H}_n.

Here is the formula. It can be proved to be concave function.