Which $A$ minimizes ${\operatorname{Tr}(A^T [(I-A)(I-A)^T]^{-1} A)}+{\operatorname{Tr}(c^T [(I-A)(I-A)^T]^{-1} c)}$ 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!
 A: 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
$$\eqalign{
 B = A-I &= -(I+cc^T) \cr
 A &= -cc^T \cr
}$$
