## Horn’s inequality for singular values via exterior algebra

$$\DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Id}{Id}$$Horn’s inequality states that for any two compact operato...

## Line integrals: 3. Applications to complex analysis

$$\newcommand{\bbc}{\mathbb{C}} \newcommand{\Int}{\operatorname{Int}}$$Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singul...

## Line integrals: 2. Locally exact forms and singular homology

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis As in part 1, $$E,F$$ a...

## Line integrals: 1. Exact, conservative and closed forms

Navigation: 1. Exact, conservative and closed forms | 2. Locally exact forms and singular homology | 3. Applications to complex analysis The line integral is a ...

## First-order ODEs, matrix exponentials, and det(exp)

Last time we derived a formula for the derivative of the matrix exponential. Here we will be focusing instead on the expression $$D\exp(x)u=\exp(x)u=u\exp(x),$$...

## Fréchet derivative of the (matrix) exponential function

$$D\exp(x)u = \int_0^1 e^{sx}ue^{(1-s)x}\,ds.$$ This intriguing formula expresses the derivative of the exponential map on a Banach algebra as an integral. In...

## Convex functions, second derivatives and Hessian matrices

In single variable calculus, a twice differentiable function $$f:(a,b)\to\mathbb{R}$$ is convex if and only if $$f^{\prime\prime}(x)\ge 0$$ for all \(x\in(a,b)\...

## Differentiation done correctly: 5. Maxima and minima

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima In this final post, we...

## Differentiation done correctly: 4. Inverse and implicit functions

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima Now we’re going ...

## Differentiation done correctly: 3. Partial derivatives

Navigation: 1. The derivative | 2. Higher derivatives | 3. Partial derivatives | 4. Inverse and implicit functions | 5. Maxima and minima While we saw that diff...