Click here
Click here
$A = U\Sigma V^{\top}$
$E = mc^2$
$F = ma$
$a^2 + b^2 = c^2$
$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}$
$e^{i\pi}+1=0$
$-\Delta u = \lambda u$
$\frac{d\mathbf{u}}{dt}=\frac{1}{\rho}\,\nabla\!\cdot\boldsymbol{\sigma}+\mathbf{a}$
Eigendeck
Eigendeck
LaTeX math & interactive technical elements … click through to see the other slides …
LaTeX math & interactive technical elements
… click through to see the other slides …
And here!
And here!
David F. Gleich · Eigendeck Showcase1
Eigenmodes of a membrane
Eigenmodes of a membrane
$-\Delta u = \lambda u$
Paint a membrane; we solve −Δu = λu on it. Pick a mode: its nodal lines (left) and the vibrating eigenvector (right). Eigenvectors of the Laplacian, made visible.
Paint a membrane; we solve −Δu = λu on it. Pick a mode: its nodal lines (left) and the vibrating eigenvector (right). Eigenvectors of the Laplacian, made visible.
Try drawing here!
Try drawing here!
David F. Gleich · Eigendeck Showcase2
Mathematics · Inequalities
Mathematics · Inequalities
Cauchy–Schwarz
Cauchy–Schwarz
$$\bigl|\langle x,\,y\rangle\bigr|\;\le\;\lVert x\rVert\,\lVert y\rVert$$
Proof For every real $t$, expand the non-negative quantity $\lVert x-t\,y\rVert^{2}=\lVert x\rVert^{2}-2t\,\langle x,y\rangle+t^{2}\lVert y\rVert^{2}\ge 0$. A quadratic in $t$ that is never negative has discriminant $\le 0$: $$\langle x,y\rangle^{2}\le\lVert x\rVert^{2}\,\lVert y\rVert^{2}.$$ ■
Proof

For every real , expand the non-negative quantity . A quadratic in that is never negative has discriminant :
■
Equality holds exactly when $x$ and $y$ are linearly dependent.
Equality holds exactly when and are linearly dependent.
David F. Gleich · Eigendeck Showcase3
Gradient descent
Gradient descent
$$f(\mathbf{x})=\tfrac12\beta\lVert\mathbf{x}\rVert^{2}-\sum_i a_i\,e^{-\lVert\mathbf{x}-\mathbf{c}_i\rVert^{2}/2s_i^{2}}$$$$\nabla f(\mathbf{x})=\beta\,\mathbf{x}+\sum_i \tfrac{a_i}{s_i^{2}}(\mathbf{x}-\mathbf{c}_i)\,e^{-\lVert\mathbf{x}-\mathbf{c}_i\rVert^{2}/2s_i^{2}}$$$$x_{k+1}=x_k-\alpha\,\nabla f(x_k)$$
Click anywhere to drop a ball — it follows the gradient into a basin on a non-convex surface.
Click anywhere to drop a ball — it follows the gradient into a basin on a non-convex surface.
David F. Gleich · Eigendeck Showcase4
Mathematics · Numerical Linear Algebra
Mathematics · Numerical Linear Algebra
The best low-rank approximation
The best low-rank approximation
Theorem · Eckart–Young–Mirsky Let $A\in\mathbb{R}^{m\times n}$ have the singular value decomposition $A=U\Sigma V^{\top}$ with $\sigma_1\ge\sigma_2\ge\cdots\ge\sigma_r\gt 0$. For every $k\lt r,$ the truncated SVD $$A_k=\sum_{i=1}^{k}\sigma_i\,u_i v_i^{\top}$$ is the closest rank-$k$ matrix to $A$ in every unitarily invariant norm, and the error is set exactly by the discarded singular values: $$\lVert A-A_k\rVert_2=\sigma_{k+1},\qquad \lVert A-A_k\rVert_F=\Bigl(\sum_{i\gt k}\sigma_i^{2}\Bigr)^{1/2}.$$
Theorem · Eckart–Young–Mirsky

Let have the singular value decomposition with . For every the truncated SVD
is the closest rank- matrix to in every unitarily invariant norm, and the error is set exactly by the discarded singular values:
David F. Gleich · Eigendeck Showcase5
Low-rank image compression
Low-rank image compression
$A\approx\sum_{k=1}^{r}\sigma_k\,u_k v_k^{\top}$
Plain SVD vs the SVD of a matrix of tiles, at equal storage. Drag the budget — the tiled reorganization wins. (Gleich, arXiv:2402.18427.)
Plain SVD vs the SVD of a matrix of tiles, at equal storage. Drag the budget — the tiled reorganization wins. (Gleich, arXiv:2402.18427.)
David F. Gleich · Eigendeck Showcase6
Looking at paths in graphs
Looking at paths in graphs
The Fauci email reply network (Benson, Veldt & Gleich, ICWSM 2022), force-directed. Click two people to highlight every shortest reply-path between them — the count and length show below. Coloured by a spectral min-cut; hover for a name; drag to re-settle.
The Fauci email reply network (Benson, Veldt & Gleich, ICWSM 2022), force-directed. Click two people to highlight every shortest reply-path between them — the count and length show below. Coloured by a spectral min-cut; hover for a name; drag to re-settle.
David F. Gleich · Eigendeck Showcase7
The wave equation
The wave equation
$\frac{\partial^2 u}{\partial t^2}=c^2\,\frac{\partial^2 u}{\partial x^2}$
Click or drag to pluck the string — the pulse splits into two waves and reflects off the ends.
Click or drag to pluck the string — the pulse splits into two waves and reflects off the ends.
David F. Gleich · Eigendeck Showcase8
Physics · Classical Electromagnetism
Physics · Classical Electromagnetism
Maxwell's equations
Maxwell's equations
Gauss's law
Gauss's law
$$\nabla\!\cdot\mathbf{E}=\dfrac{\rho}{\varepsilon_0}$$
No magnetic monopoles
No magnetic monopoles
$$\nabla\!\cdot\mathbf{B}=0$$
Faraday — induction
Faraday — induction
$$\nabla\times\mathbf{E}=-\dfrac{\partial\mathbf{B}}{\partial t}$$
Ampère–Maxwell
Ampère–Maxwell
$$\nabla\times\mathbf{B}=\mu_0\mathbf{J}+\mu_0\varepsilon_0\dfrac{\partial\mathbf{E}}{\partial t}$$
In vacuum the fields obey $\partial_t^2\mathbf{E}=c^2\nabla^2\mathbf{E}$ — a wave at speed $c=1/\sqrt{\mu_0\varepsilon_0}$. Light is electromagnetism.
In vacuum the fields obey — a wave at speed . Light is electromagnetism.
David F. Gleich · Eigendeck Showcase9
Fourier & the FFT
Fourier & the FFT
$X_k=\sum_{n=0}^{N-1} x_n\,e^{-2\pi i kn/N}$
A signal and its frequency spectrum. Drag "keep" to rebuild it from only its largest Fourier coefficients — or draw your own. Real FFT, live.
A signal and its frequency spectrum. Drag "keep" to rebuild it from only its largest Fourier coefficients — or draw your own. Real FFT, live.
David F. Gleich · Eigendeck Showcase10
Polynomial interpolation
Polynomial interpolation
The Runge phenomenon: interpolating through uniform nodes blows up near the ends as the degree grows, while Chebyshev nodes (clustered at the ends) converge. Drag the degree.
The Runge phenomenon: interpolating through uniform nodes blows up near the ends as the degree grows, while Chebyshev nodes (clustered at the ends) converge. Drag the degree.
David F. Gleich · Eigendeck Showcase11
“The purpose of computing is insight, not numbers.”
“The purpose of computing is insight, not numbers.”
— Richard W. Hamming
— Richard W. Hamming
David F. Gleich · Eigendeck Showcase12
Molecule viewer
Molecule viewer
Pick a molecule from the list and drag to rotate the 3-D ball-and-stick model — from water up to a C₆₀ buckyball. Computed geometry.
Pick a molecule from the list and drag to rotate the 3-D ball-and-stick model — from water up to a C₆₀ buckyball. Computed geometry.
David F. Gleich · Eigendeck Showcase13
Neural network
Neural network
A small ResNet classifying MNIST — draw a digit and watch it flow through the layers. Real trained weights, live inference.
A small ResNet classifying MNIST — draw a digit and watch it flow through the layers. Real trained weights, live inference.
David F. Gleich · Eigendeck Showcase14
Computer Science · Analysis of Algorithms
Computer Science · Analysis of Algorithms
The Master Theorem
The Master Theorem
For a divide-and-conquer recurrence with $a\ge 1$ subproblems of size $n/b$ (here $b\gt 1$), set $c=\log_b a$ and compare $f(n)$ with $n^{c}$: $$T(n)=a\,T\!\left(\frac{n}{b}\right)+f(n)$$
For a divide-and-conquer recurrence with subproblems of size (here ), set and compare with :
1 · Leaves dominate If $f(n)=O\!\left(n^{c-\epsilon}\right)$, $$T(n)=\Theta\!\left(n^{c}\right)$$
1 · Leaves dominate

If ,
2 · Balanced If $f(n)=\Theta\!\left(n^{c}\right)$, $$T(n)=\Theta\!\left(n^{c}\log n\right)$$
2 · Balanced

If ,
3 · Root dominates If $f(n)=\Omega\!\left(n^{c+\epsilon}\right)$ & $a\,f(n/b)\le\kappa f(n)$, $$T(n)=\Theta\!\left(f(n)\right)$$
3 · Root dominates

If & ,
merge sort: T(n) = 2·T(n/2) + Θ(n) ⇒ Θ(n log n)
merge sort: T(n) = 2·T(n/2) + Θ(n) ⇒ Θ(n log n)
David F. Gleich · Eigendeck Showcase15
$A = U\Sigma V^{\top}$
$E = mc^2$
$F = ma$
$a^2 + b^2 = c^2$
$\int_{-\infty}^{\infty} e^{-x^2}\,dx=\sqrt{\pi}$
$e^{i\pi}+1=0$
$-\Delta u = \lambda u$
$\frac{d\mathbf{u}}{dt}=\frac{1}{\rho}\,\nabla\!\cdot\boldsymbol{\sigma}+\mathbf{a}$
Eigendeck
Eigendeck
We can’t wait to see what you use this to do! eigendeck.dev
We can’t wait to see what you use this to do!
eigendeck.dev
David F. Gleich · Eigendeck Showcase16