Suppose we have a typical logdet function $\mathcal{L}$ 
$$
\mathcal{L} = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mathbf{q}$ is a column vector, $\mathbf{I}$ is an identity matrix. The local minima of $\mathcal{L}$ w.r.t $\mathbf{A}$ could be found by solving $\mathbf{A}$ ($\frac{\partial \mathcal{L}}{\partial \mathbf{A}} = \mathbf{0}$), we get the closed form solution of $\mathbf{A}$
$$
\mathbf{A} = \mathbf{S}^{-1}(\mathbf{S} - \mathbf{q}\mathbf{q}^T)\mathbf{S}^{-1}
$$


**The problem is**, what if I want to constrain the descent path of the logdet function by imposing extra structures on $\mathbf{A}$ ? For example,


1. $\mathbf{A}$ is p.s.d and have a simple structure, which constrains that at each step, $\mathbf{A}$ should be: $\mathbf{A}=r\mathbf{I}$.
2. or we constrain that $\mathbf{A}$ is p.s.d and have a band-p Symmetric Toeplitz structure, i.e.,
\begin{equation}
\mathbf{A} = r 
\begin{pmatrix}
1 & p & 0 & 0 \\\
p & 1 & p & 0 \\\
0 & p & 1 & p \\\
0 & 0 & p & 1 
\end{pmatrix}.
\end{equation}


How could I analyze the saddle point of $\mathcal{L}$ with these constrains ? Will these constrains affect achieving the local minima of $\mathcal{L}$ and provide a more robust result that more likely to find the true minimal that has **this** type of structure ?



Regularized Gradient with respect to a matrix (with a specific structure)
@liubenyuan, some remarks. Necessarily $S>0$. On the other hand, your result is correct up a signum (have a look at the case $n=1$); infortunately that makes a big difference ! The derivative is $0$ if $A=S^{-1}(qq^T-S)S^{-1}$ and you want that $A\geq 0$, that is $qq^T\geq S$. Let $(\lambda\_i\geq 0)\_i$ be the spectrum of $S$. According to the min-max theorem $||q||^2\geq \lambda\_1$ and, for every $i>1$, $\lambda\_i=0$ and $S$ cannot be invertible except if $n=1$. Finally, there are no critical points, except for $n=1$, $a=q^2/s^2-1/s$ with $q^2\geq s$.


EDIT: cf. the following post. Have you read my answer to this post ?


[Hessian of function of covariance matrices](https://mathoverflow.net/questions/133862/hessian-of-function-of-covariance-matrices)


