To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the intuition behind the claim that, if the Hessian $H$ of a multidimensional differentiable function $f(x_1,…,x_n)$ is positive semi-definite, it must be […]

If $\mathbf M=\left[\begin{matrix}\bf A & \bf b\\\bf b^\top & \bf d \\\end{matrix}\right]$ such that $\bf A$ is positive definite, under what conditions is $\bf M$ positive definite, positive semidefinite and indefinite? It is readily seen that $\det(\mathbf M)=\alpha\det(\mathbf A)$, where $\alpha=\mathbf d-\bf b^\top A^{-1}b$ Now, $\alpha>0\Rightarrow \det(\mathbf M)>0$ $\quad(\det(\mathbf A)>0$ by hypothesis$)$ This is not […]

Let $A\in\mathbb C^{n\times n}$ be a Hermitian matrix such that all its principal minors are non-negative (i.e. for $B=\left(a_{l_il_j}\right)_{1≤i,j≤k}$ with $1≤l_1<…<l_k≤n$ we have $\det(B)≥0$). Then how to show that $A$ is positive semi-definite? I thought maybe we could use induction since the condition is also satisfied for every submatrix, but I couldn’t find an easy […]

There is a theorem which states that every positive semidefinite matrix only has eigenvalues $\ge0$ How can I prove this theorem?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? My proof of the positive definite case falls apart for the semidefinite case because of the possibility of division by zero…

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le \lambda f(x) + (1- \lambda)f(y)$ for any $x,y \in \mathbb{R}^2$. Hessian matrix is nonnegative-definite if $f_{xx}” x^2 + f_{x,y}(x+y) + f_{yy}”y^2 \ge 0$ I know […]

Intereting Posts

Proof if $I+AB$ invertible then $I+BA$ invertible and $(I+BA)^{-1}=I-B(I+AB)^{-1}A$
Is a function whose derivative vanishes at rationals constant?
Examples of patterns that eventually fail
This tower of fields is being ridiculous
Let $p$ be prime and $(\frac{-3}p)=1$. Prove that $p$ is of the form $p=a^2+3b^2$
Is there a formula for finding the number of nonisomorphic simple graphs that have n nodes?
How (and why) would I reparameterize a curve in terms of arclength?
How to learn from proofs?
Derived category and so on
Prove that $\angle FGH = \angle GDJ$
Ramanujan's 'well known' integral, $\int_\frac{-\pi}{2}^\frac{\pi}{2} (\cos x)^m e^{in x}dx$.
Distribution of compound Poisson process
Let $K$ be a field and $f(x)\in K$ be a polynomial of degree $n$. And let $F$ be its splitting field. Show that $$ divides $n!$.
Solving higher order logarithms integrals without the beta function
Example where $f\circ g$ is bijective, but neither $f$ nor $g$ is bijective