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...
Category: Mathematics
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...