minimize the trace of a matrix function

Question

G,H are $n \times n$ Positive definite symmetric matrix and $LGL^{T}=H$, L is $n \times n$ as well, find the L to minimize the $$g(L)=tr((I-L)G(I-L)^{T})$$ I learned how to calculate the max of trace by the inner product property, but I feel confused on the minimize problem, I can not find a proper tool to deal with such problem. I notice that there are some derivation method on matrix, is that the tool that I need? Can you give me some chapters of books to help me to learn?

Answer 1

Let $X = H^{-1/2}LG^{1/2}$ (correspondingly, $L = H^{1/2}XG^{-1/2}$). Then $XX^\mathsf{T} = I$, i.e., $X$ is orthogonal. We have \begin{align} g(L) &= \mathrm{Tr}(G - LG - GL^\mathsf{T} + LGL^\mathsf{T})\\ &= \mathrm{Tr}(G + H) - 2 \mathrm{Tr}(LG)\\ &= \mathrm{Tr}(G + H) - 2 \mathrm{Tr}(G^{1/2}H^{1/2}X)\\ &= \mathrm{Tr}(G + H) - 2 \mathrm{Tr}(UDV^TX)\\ &= \mathrm{Tr}(G + H) - 2 \mathrm{Tr}(DY)\\ &\ge \mathrm{Tr}(G + H) - 2 \mathrm{Tr}(D) \end{align} with equality if $Y = I$; here, $UDV^T$ is the singular value decomposition of $G^{1/2}H^{1/2}$ ($U$ and $V$ are orthogonal, $D$ is diagonal), and $Y = V^TXU$ (so $Y$ is orthogonal). As a result, the minimum of $g(L)$ is $\mathrm{Tr}(G + H) - 2 \mathrm{Tr}(D)$ achieved at $L = H^{1/2}VU^\mathsf{T}G^{-1/2}$.