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A linear equation is an equation in which the unknowns occur only to the first power and are combined linearly.

Starting at the same value, forward and backward Euler have opposite leading local errors. Their half-difference therefore estimates either local error:For a first-order method, a local-tolerance controller consequently scales the next step by $(\mathrm{tol}/E_n{)}^{1/2}$, usually with a safety factor.

If a linear system has negative eigenvalues with widely separated magnitudes, an explicit method can be forced to resolve the fastest decaying mode solely for stability. For eigenvalues $-1$ and $-100$, forward Euler requires $h\le 0.02$, whereas backward Euler is stable for every positive step size.

A method is A-stable when its stability domain contains the closed left half-plane.

For a real $2\times 2$ Jacobian matrix $J$, both eigenvalue have negative real part exactly whenThe linearization stability theorem then makes a hyperbolic equilibrium locally asymptotically stable.

A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.

This criterion concerns travelling-front solutions of Fisher's equation.

For diffusion coefficient $D$ and leading-edge growth rate $r>0$, a Fisher-KPP-type pulled front has minimum speed $2\sqrt{Dr}$.

The transportation problem minimizes linear shipping cost while matching prescribed row supplies and column demands.

This is the finite discrete optimal transport problem with prescribed supply and demand marginals.

At a primal-dual optimum, each positive variable corresponds to a tight dual constraint and conversely for positive dual variables.

Every feasible dual objective value bounds every feasible primal objective value.

The dual pairsatisfies weak duality because $b^{T}y\le y^{T}Ax=x^T{A}^{T}y\le c^{T}x$ for every feasible pair.

For a primal maximization problem $Ax\le b$, $x\ge 0$, the dual minimizes $b^{T}y$ subject to $A^{T}y\ge c$, $y\ge 0$.

Rational canonical form expresses a linear map as a direct sum of companion matrices of invariant factors, and works over any field.

An invariant direct-sum decomposition splits a space into nonzero subspaces preserved by the operator.

A companion matrix represents multiplication by x in the power basis modulo a monic polynomial.

A cyclic vector has iterates under an operator that span the whole vector space.

The Rayleigh quotient measures the quadratic form of a self-adjoint operator per unit squared norm. For a nonzero vector $x$ and a finite-dimensional Hermitian matrix $A$,For a real symmetric matrix, replace the conjugate transpose $x^{\ast }$ by $x^T$. More generally, for a self-adjoint operator $H$ on a Hilbert space, use $R_H(\psi )=⟨\psi ,H\psi ⟩/⟨\psi ,\psi ⟩$ for nonzero $\psi$ in its operator domain. On an eigenvector it returns the exact eigenvalue. Multiplying the vector by any nonzero scalar leaves the quotient unchanged, so it measures a direction rather than a chosen length.

For the symmetric matrix $A=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ from the finite-dimensional spectral theorem example,The first value is not an eigenvalue: it is an average of the two directional responses. To see every direction, write the unit vector $q(\theta )=\cos \theta e_{+}+\sin \theta e_{-}$ in the orthonormal eigenbasis. ThenThe largest value is $3$ along $e_{+}$; the smallest is $1$ along $e_{-}$. A direction halfway between them has quotient $2$. In the left panel, the color on the unit circle represents the quotient; in the right panel, the same values are plotted against the angle measured from $e_{+}$.

The Hermitian Rayleigh quotient range explains why this is useful: evaluating one trial vector gives an upper bound on the smallest eigenvalue and a lower bound on the largest one, without solving a characteristic polynomial. At an extreme eigenvector, a small angular error changes the quotient only quadratically. Near $e_{+}$ in this example, $3-R_A(q(\theta ))=2{\sin }^2\theta =2{\theta }^2+O({\theta }^4)$. This is the geometric reason for quadratic Rayleigh-quotient improvement for the power method and motivates the Rayleigh-Ritz method.

There is a concrete historical link with normal modes. In The Theory of Sound (1877), Volume I, Sections 88–89, Lord Rayleigh studies the period of a constrained trial vibration. He shows that the period has a stationary point at a normal mode and that a trial type bounds the longest natural period. These energy ratios are a mechanical setting for the modern Rayleigh quotient and generalized Rayleigh quotient; the book describes the result through vibration periods, rather than today's matrix notation.^[ref]

The unilateral shifts on sequences move every component one place left or right. Their spectra illustrate behavior absent in finite dimensions.

A matrix is normal when $A{A}^{†}=A^{†}A$.

A complex matrix is unitary when $U^{†}U=U{U}^{†}=I$. Its columns form an orthonormal basis, and it preserves inner products and norms.