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A permutation matrix has exactly one entry equal to one in each row and column and zeros elsewhere. Left or right multiplication permutes coordinates, and $P^{-1}=P^T$.

An upper Hessenberg matrix has zero entries below its first subdiagonal: $a_{ij}=0$ whenever $i>j+1$.

Matrix multiplication represents composition of linear maps. Its entries are $(AB{)}_{ij}={\sum }_k{A}_{ik}{B}_{kj}$.

An upper bidiagonal matrix can have nonzero entries only on its main diagonal and first superdiagonal.

A nonzero matrix has rank one exactly when it can be written as an outer product $u{v}^T$.

The identity matrix has ones on its main diagonal and zeros elsewhere, and satisfies $IA=AI=A$ whenever the products are defined.

The rank of a matrix is the dimension of its column space, equivalently the dimension of its row space.

The matrix unit $E_{ij}$ has entry one in row $i$, column $j$, and zero in every other position. The matrix units form the standard basis of the vector space of matrices of a fixed size.

A matrix element $A_{ij}$ is the entry in row $i$ and column $j$ of a matrix $A$.

For a square matrix $A$ and a nonnegative integer $k$, the matrix power is the repeated productwith $A^0=I$.

A switchback term such as $ϵ\log ϵ$ is forced into a matched expansion when logarithms of a stretched coordinate split into parameter-dependent and local pieces in the overlap region.

An outer expansion describes the solution away from a thin or distant region resolved by an inner expansion. It is matched to that inner expansion in their common overlap region. Its range of validity must be specified: an approximation at each fixed positive $x$ need not have uniform convergence down to an endpoint.

For the singular perturbation problem $ϵy^{\prime \prime }+y^{\prime }=0$, $y(0)=0$, $y(1)=1$, try the asymptotic expansion $y\sim y_0+ϵy_1+{ϵ}^2{y}_2+\cdots$ at fixed $x>0$. Substitution gives $y_0^{\prime }=0$ and $y_n^{\prime }=-y_{n-1}^{\prime \prime }$ for $n\ge 1$. Expanding the right boundary condition gives $y_0(1)=1$ and $y_n(1)=0$ for $n\ge 1$. Therefore $y_0=1$ and every higher coefficient vanishes. The missing boundary layer is exponentially small at each fixed positive $x$, invisible to all these powers of $ϵ$.

The exact solution supplies a quantitative limit. For every fixed $a\in (0,1)$,This is uniform convergence on $[a,1]$. On $[0,1]$ the same supremum norm of the error is exactly one. The inner expansion supplies the endpoint information that this outer expansion cannot encode.

An inner expansion resolves a region in which a stretched independent variable makes otherwise small effects enter the leading balance.

If ${\mathcal{F}}_n$ is generated by independent symmetric signs ${\xi }_1,\dots ,{\xi }_n$, every martingale has predictable coefficients $B_n$ such thatOne may take $B_n=\mathbb{E}[M_n{\xi }_n\mid {\mathcal{F}}_{n-1}]$.

Let $S_0=0$ be a simple symmetric random walk, let $T_a$ be its first hitting time of the positive integer $a$, and reverse every increment after $T_a$. The resulting path is again a simple symmetric random walk. Consequently,

The continuous analogue is the existing Reflection principle (Wiener process); its tail formula has no discrete one-unit lattice correction.

For a stopping time $T$,The indicator is predictable, so a stopped integrable martingale is a martingale.

For a square-integrable martingale transform and positive conditional increment variance,

A right-continuous continuous-time martingale with ${\sup }_{t\ge 0}\mathbb{E}|M_t|<\infty$ has an almost-sure finite limit as $t\to \infty$.

A Majorana zero mode is a spatially localized Majorana operator that commutes with a gapped quadratic Hamiltonian, exactly or up to exponentially small finite-size corrections.

For a vertical magnetic field in a Keplerian disk, axisymmetric vertical MRI modes are unstable when $k^2{v}_A^2<3{\Omega }^2$. The shortest unstable wavelength is ${\lambda }_{\mathrm{crit}}=2\pi v_A/(\sqrt{3}\Omega )$.