FOURIER FIELD

\hat{f} (ω) = \int_{- \infty}^{\infty} f (x) e^{- iω x} d x

f (x) = \frac{1}{2 π} \int \hat{f} (ω) e^{iω x} d ω

PROJECTION

P = Q Q^{⊤}

r = b - Pb ⊥ col (Q)

GREEN LOOP

\oint_{\partial S} - y d x + x d y = 2 A

\iint_{S} (\nabla \times F) \cdot d S

NORM GEOMETRY

∥ v ∥_{S}^{2} = v^{⊤} Sv

∥ A ∥_{2} = ∥ x ∥_{2} = 1 max ∥ Ax ∥_{2}

RANK NULLITY

rank (A) + dim ker A = n

Ax = 0

LAPLACE MODE

L {f} (s) = \int_{0}^{\infty} f (t) e^{- s t} d t

L {f^{'}} = s F (s) - f (0)

STOKES RIM

\iint_{S} (\nabla \times F) \cdot d S = \oint_{\partial S} F \cdot d r

\partial S \to S

LAURENT BAND

f (z) = n = - \infty \sum \infty a_{n} (z - z_{0})^{n}

a_{n} = \frac{1}{2 π i} \oint \frac{f ( z )}{( z - z _{0} ) ^{n + 1}} d z

TENSOR TRACE

(AB)_{ij} = k \sum A_{ik} B_{k j}

tr (A) = i \sum A_{ii}

TIKHONOV

x min ∥ Ax - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}

x_{λ} = (A^{⊤} A + λ I)^{- 1} A^{⊤} b

SHERMAN MORRISON

(A + u v^{⊤})^{- 1} = A^{- 1} - \frac{A ^{- 1} u v ^{⊤} A ^{- 1}}{1 + v ^{⊤} A ^{- 1} u}

Δ A = u v^{⊤}

SPARSE CODE

x = Ds, ∥ s ∥_{0} ≪ n

s min ∥ y - Ds ∥_{2}^{2}

CONDITION NUMBER

κ_{2} (A) = \frac{σ _{m a x}}{σ _{m i n}}

\frac{∥ δ x ∥}{∥ x ∥} ≲ κ (A) \frac{∥ δ b ∥}{∥ b ∥}

ORTHOGONAL MAP

Q^{⊤} Q = I

∥ Qx ∥_{2} = ∥ x ∥_{2}

FINITE DIFFERENCE

\frac{u _{i + 1} - 2 u _{i} + u _{i - 1}}{h ^{2}} \approx u^{''} (x_{i})

\partial_{x} u \approx \frac{u _{i + 1} - u _{i - 1}}{2 h}

SVD SKETCH

X \approx U_{r} Σ_{r} V_{r}^{⊤}

∥ X - X_{r} ∥_{2} = σ_{r + 1}

BAYES UPDATE

P (A ∣ B) = \frac{P ( B ∣ A ) P ( A )}{P ( B )}

P (B) = i \sum P (B ∣ A_{i}) P (A_{i})

HEAT TRACE

\partial_{t} u = α Δ u

u (x, t) = k \sum c_{k} e^{- α λ_{k} t} ϕ_{k} (x)

EIGEN FLOW

A = SΛ S^{- 1}

e^{t A} = S e^{t Λ} S^{- 1}

KRYLOV SPACE

K_{m} (A, r_{0}) = span {r_{0}, A r_{0}, \dots}

x_{m} \in x_{0} + K_{m}

HOUSEHOLDER

H = I - 2 v v^{⊤}

H^{⊤} H = I

MARKOV STEP

p_{k + 1} = P p_{k}

1^{⊤} P = 1^{⊤}

WAVE MODE

u_{tt} = c^{2} Δ u

\overset{u}{^}_{tt} = - c^{2} ∥ k ∥^{2} \overset{u}{^}

GAUSS FLUX

∭_{V} \nabla \cdot F d V = \iint_{\partial V} F \cdot n d S

Φ = F \cdot n

RESIDUE

\oint_{γ} f (z) d z = 2 π i k \sum Res (f, z_{k})

a_{- 1} = Res (f, z_{0})

DE MOIVRE

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

z^{n} = r^{n} e^{in θ}

HESSIAN

H_{ij} = \frac{\partial ^{2} f}{\partial x _{i} \partial x _{j}}

δ f \approx \nabla f^{⊤} δ x + \frac{1}{2} δ x^{⊤} H δ x

KALMAN GAIN

K = P H^{⊤} (HP H^{⊤} + R)^{- 1}

\hat{x}^{+} = \hat{x}^{-} + Kr

RLS UPDATE

P_{k} = λ^{- 1} (P_{k - 1} - K_{k} x_{k}^{⊤} P_{k - 1})

K_{k} = \frac{P _{k - 1} x _{k}}{λ + x _{k}^{⊤} P _{k - 1} x _{k}}

VANDERMONDE

V_{ij} = λ_{j}^{i - 1}

x (0) = Vc

CHOLESKY

S = L L^{⊤}

x^{⊤} Sx > 0

GRAM SCHMIDT

q_{k} = \frac{a _{k} - \sum _{j < k} ( q _{j}^{⊤} a _{k} ) q _{j}}{∥ \cdot ∥ _{2}}

span {q_{j}} = span {a_{j}}

RK4 STEP

x_{n + 1} = x_{n} + \frac{h}{6} (k_{1} + 2 k_{2} + 2 k_{3} + k_{4})

k_{1} = f (t_{n}, x_{n})

LEAST SQUARES

x min ∥ Ax - b ∥_{2}^{2}

x^{⋆} = (A^{⊤} A)^{†} A^{⊤} b

PDE FLOW

\nabla \cdot u = 0

\partial_{t} u + \nabla p = ν Δ u

QUADRATIC FORM

q (x) = x^{⊤} Sx

\nabla_{x} q = (S + S^{⊤}) x

COMPRESSED SENSING

x min ∥ x ∥_{1}

s.t. Φx = y

QR FACTOR

A = QR

Q^{⊤} Q = I

PCA AXIS

C = \frac{1}{m} X X^{⊤}

C u_{k} = λ_{k} u_{k}

DFT GRID

X_{k} = n = 0 \sum N - 1 x_{n} e^{- i 2 π k n / N}

F^{- 1} = \frac{1}{N} \overline{F}

POISSON FIELD

- Δ u = f

\nabla \cdot \nabla u = Δ u

CAUCHY RIEMANN

u_{x} = v_{y}, u_{y} = - v_{x}

f^{'} (z) = u_{x} + i v_{x}

EULER SPIRAL

e^{i θ} = cos θ + i sin θ

z = r e^{i θ}

JACOBIAN

J_{ij} = \frac{\partial F _{i}}{\partial x _{j}}

d y = J d x

GRADIENT STEP

x_{k + 1} = x_{k} - η \nabla f (x_{k})

\nabla f = 0

WOODBURY

(A + UCV)^{- 1} = A^{- 1} - A^{- 1} U M^{- 1} V A^{- 1}

M = C^{- 1} + V A^{- 1} U

INCOHERENCE

μ = n i, j max ∣ ⟨ ϕ_{i}, ψ_{j} ⟩ ∣

y = ΦΨs

LU PIVOT

PA = LU

det A = det P i \prod U_{ii}

PSEUDOINVERSE

A^{†} = V Σ^{†} U^{⊤}

A A^{†} b = proj_{col (A)} b

DETERMINANT VOLUME

Vol = ∣ det A ∣

det (AB) = det A det B

WIR MÜSSEN WISSEN. WIR WERDEN WISSEN.

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