# Questions tagged [hessian-matrix]

The Hessian matrix of function is used to second derivative test when $f$ has a critical point $x$. If the Hessian is positive definite at $x$, then $f$ attains a local minimum at $x$. If the Hessian is negative definite at $x$, then $f$ attains a local maximum at $x$. If the Hessian has both positive and negative eigenvalues then $x$ is a saddle point for $f$.

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### Eigenvalue Optimization, Find indefinite matrix with only one negative eigenvalue

I am new to the field of eigenvalue optimization. Say I have a symmetric matrices $A(x)\in\mathbb{R}^{2n\times2n}$ which depend on $x\in\mathbb{R}^n$. I now want to find situations in which one and ...
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### Is this multivariate function convex? And is a Hessian matrix of this form always convex?

I'm looking at the following function: $$K(Q, s) = h\frac{(Q-s)^2}{2Q}+b\frac{s^2}{2Q}+K\frac{\lambda}{Q}+c\lambda,$$ where $Q>0$, $s \ge 0$ and $h,b,c, \lambda, K \ge 0$ and are known. I want to ...
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### Hessian matrix of $\Lambda \mapsto y' (I + X\Lambda X')^{-1}y$

I have $$f=y'M^{-1}y$$ where $$M = I + X\Lambda X'$$ for $y \in \mathbb{R}^n$, $X\in \mathbb{R}^{n\times p}$, and $\Lambda$ is a $p\times p$ symmetric positive-definite matrix. I'm trying to compute ...
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### 2nd order EKF covariance propagation equation (hessian)

I am trying to implement a 2nd-order EKF and am having some issues with the propagation equation for covariance. From the literature, if $X$ has covariance $P$ and the propagation function $f$ has ...