Gradient of $\log \left( \det \left( \dfrac {3}{10}I+xx^{T}\right) \right) $

Let $M = \left( xx^T + 0.3I \right)$ and $f = \log \det \left( M \right)$.

We will utilize the following the identities

• Trace and Frobenius product relation $$A:B={\rm tr}(A^TB)$$ or $$A^T:B={\rm tr}(AB)$$
• Cyclic property of Trace/Frobenius product $$\eqalign{ A:BC &= AC^T:B \cr &= B^TA:C \cr &= {\text etc.} \cr }$$
• Jacobi's formula (for nonsingular matrix $M$) in terms of differential $$d\log \det \left( M \right) = d{\rm tr}\log\left( M \right) .$$

Now, we obtain the differential first and thereafter we obtain the gradient.

So, \begin{align} df &= d \log \det \left( M \right) \\ &= d \ {\rm tr}\left( \log\left( M \right) \right) \hspace{8mm} \text{note: utilized Jacobi's formula} \\ &= {\rm tr} \left( M^{-1} dM \right) \\ &= M^{-T} \ : \ dM \hspace{8mm} \text{note: utilized trace and Frobenius relation} \\ &=M^{-1}\ : \ \left( dxx^T + xdx^T\right)\\ &= 2M^{-1}x \ : \ dx \\ \end{align}

So, the derivative of $f = \log \det \left( xx^T + 0.3I \right)$ with respect to $x$ is \begin{align} \frac{\partial}{\partial x} f = \frac{\partial}{\partial x} \log \det \left( xx^T + 0.3I \right) = 2 \left(xx^T + 0.3I\right)^{-1} x .\\ \end{align}

Let function $f : \mathbb{R}^n \to \mathbb{R}$ be defined by

$$f (\mathrm x) := \log \left( \det \left( \gamma \, \mathrm I_n + \mathrm x \mathrm x^\top \right) \right)$$

where $\gamma = \frac{3}{10}$. Using the matrix determinant lemma,

$$\det \left( \gamma \, \mathrm I_n + \mathrm x \mathrm x^\top \right) = \cdots = \gamma^n \left( 1 + \frac{1}{\gamma} \mathrm x^\top \mathrm x \right)$$

and, thus,

$$f (\mathrm x) = n \log (\gamma) + \log \left( 1 + \frac{1}{\gamma} \mathrm x^\top \mathrm x \right)$$

Taking the partial derivative with respect to $x_i$,

$$\partial_{x_i} f (\mathrm x) = \frac{\frac{2}{\gamma} x_i}{1 + \frac{1}{\gamma} \mathrm x^\top \mathrm x} = \left(\frac{2}{\gamma + \mathrm x^\top \mathrm x}\right) x_i$$

and, hence, the gradient of $f$ is

$$\nabla f (\mathrm x) = \color{blue}{\left(\frac{2}{\gamma + \mathrm x^\top \mathrm x}\right) \mathrm x}$$