Given that $A$ is a strictly lower triangular matrix and $c$ a column vector with $c^Tc=k$, where $k$ is a constant scalar, is it possible to come up with an analytical expression for $A$ which minimizes following expressions:
$$ \min_A{\operatorname{Tr}(A^T [(I-A)(I-A)^T]^{-1} A)+{\operatorname{Tr}(c^T [(I-A)(I-A)^T]^{-1} c)}} $$
Many thanks in advance!
Note that the double-contraction product is a convenient infix notation (:) for the trace, i.e. $$A:B={\rm tr}(A^TB)$$ It has nice properties due to the cyclical properties of the trace. Let me list a few $$\eqalign{ A:B &= B:A = B^T:A^T \cr B^TA:C &= A:BC = AC^T:B \cr }$$ The first thing we need is a change of variables. $$B=(A-I) \implies A=(B+I)$$ Then we can write the objective function in terms of this new variable and find its differential & gradient $$\eqalign{ f &= A:(BB^T)^{-1}A + c:(BB^T)^{-1}c \cr &= (B+I):(BB^T)^{-1}(B+I) + c:(BB^T)^{-1}c \cr &= B:(BB^T)^{-1}B + B:(BB^T)^{-1} + I:(BB^T)^{-1}B + I:(BB^T)^{-1} + c:(BB^T)^{-1}c \cr &= (BB^T):(BB^T)^{-1} + B:B^{-T}B^{-1} + I:B^{-T}B^{-1}B + I:B^{-T}B^{-1} + c:B^{-T}B^{-1}c \cr &= {\rm rank}(BB^T) + 2I:B^{-1} + B^{-1}:B^{-1} + B^{-1}c:B^{-1}c \cr \cr df &= 0 + 2I:dB^{-1} + 2B^{-1}:dB^{-1} + 2B^{-1}c:dB^{-1}c \cr &= 2\Big(I + B^{-1} + B^{-1}cc^T\Big):dB^{-1} \cr &= -2B^{-T}\Big(I + B^{-1} + B^{-1}cc^T\Big)B^{-T}:dB \cr &= -2B^{-T}\Big(I + B^{-1}(I+cc^T)\Big)B^{-T}:dB \cr \cr \frac{\partial f}{\partial B} &= -2B^{-T}\Big(I + B^{-1}(I+cc^T)\Big)B^{-T} \cr }$$ Setting the gradient to zero yields an equation which can be solved for $B$ $$\eqalign{ I &= -B^{-1}(I+cc^T) \cr B &= -(I+cc^T) \cr }$$ And finally, we can switch back to the original variable $$\eqalign{ B = A-I &= -(I+cc^T) \cr A &= -cc^T \cr }$$