# Linear Algebra Question concerning the trace of a symmetric positive definite matrix.

The objective is to minimize the diagonal elements of a symmetric positive definite matrix. The expression of this matrix is a little bit nasty and its inverse is much easier to deal with.
Can I claim that minimizing the diagonal entries of this matrix is equivalent to maximizing those of its inverse ?

If it helps, the matrix is of the form:
$A = B - BC^{T}\Lambda^{-1}CB$.
Its inverse is of the following form:
$A^{-1}=B^{-1}+kC^{T}C$. For some known "$k$".
$B$ is a toeplitz symmetric positive definite matrix.
$\Lambda$ is a symmetric positive definite matrix.
$C$ is a $1\times2$ block matrix where the first part is diagonal and the second block is the all zero matrix. (To match the dimensions).

For symmetric positive definite $A$, $\exists$ $D$ with $D_{ii}\neq 0$ s.t. $A=PDP^{-1}$ and $A^{-1}=PD^{-1}P^{-1}$ where $D_{ii}\cdot(D^{-1})_{ii}=1$. So yes, minimising $A$ is equivalent to maximising $A^{-1}$.