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For an approximate eigenpair $(\tilde{\lambda },\tilde{v})$, the residual is

An LDL decomposition of a real symmetric matrix iswhere $L$ is unit lower triangular and $D$ is diagonal. Symmetric elimination computes it one Schur complement at a time. The matrix is positive definite exactly when every diagonal pivot in $D$ is positive.

Gaussian elimination applies elementary row operations to reduce a linear system to triangular form. Dense elimination uses $O(n^3)$ arithmetic operations.

Gaussian elimination solves a linear system by removing one unknown from later equations at each step. Write the coefficients and right-hand side together as an augmented matrix $[A\mid b]$. Apply the same elementary row operations to both parts until the coefficients form a row echelon form; for a square invertible matrix, this gives an upper triangular matrix system. Then back substitution recovers the unknowns from the last equation upward. The useful simplification is that each successive equation involves fewer unknowns, while the set of solutions stays the same.

Why do elementary row operations preserve solutions? Interchanging equations changes only their order. Multiplying an equation by a nonzero scalar can be undone by division. Replacing equation $i$ by itself plus $c$ times equation $j$, with $i\ne j$, can be undone by adding $-c$ times the unchanged equation $j$. Equivalently, an elementary matrix $E$ is an invertible matrix, so $Ax=b$ if and only if $EAx=Eb$. Applying an operation to $A$ without also applying it to $b$ loses this equivalence.

At step $k$, choose a nonzero pivot element $a_{kk}$ after any row interchange. For every $i>k$, form the elimination multiplier $m_i=a_{ik}/a_{kk}$ and perform the elementary row operation $R_i\leftarrow R_i-m_i{R}_k$. Its coefficient in column $k$ becomes $a_{ik}-(a_{ik}/a_{kk})a_{kk}=0$ in exact arithmetic. Later operations combine rows that already have zeros in earlier columns, so those zeros survive. If an active column has no nonzero candidate, general row echelon form reduction skips it; a square invertible matrix always has a candidate at each step with suitable row interchanges. Partial pivoting also considers roundoff error when selecting the candidate.^[ref]

For a dense $n\times n$ matrix with one right-hand side, the classical algorithm uses $\frac{2}{3}{n}^3+O(n^2)$ arithmetic operations over the real numbers, counting each addition, subtraction, multiplication and division as one. At step $k$, the coefficient update has $(n-k{)}^2$ entries, each costing one multiplication and one subtraction; the elimination multipliers, right-hand-side updates and back substitution contribute lower-order terms. Searching for partial pivoting adds $O(n^2)$ comparisons, and storing the dense coefficients uses $O(n^2)$ space. These assumptions exclude the different storage and work patterns of sparse Gaussian elimination; the exact loop counts appear in LU factorization and solve costs. Here $O$ denotes Big O notation.

Recording the same steps as an LU decomposition makes Reusing LU factors for multiple right-hand sides possible. This is useful whenever the coefficients stay fixed while the forcing vector changes: the expensive elimination is done once. The choice of pivot element matters even when the linear system is well conditioned, as Tiny-pivot roundoff in Gaussian elimination demonstrates.

For an affinely parametrized null congruence in $D$ spacetime dimensions,

Given null vectors $k$ and $n$ with $k\cdot n=-1$, the screen-space projector isIt annihilates $k$ and $n$ and, in $D$ spacetime dimensions, projects onto the $(D-2)$-dimensional spacelike space transverse to them.

ChooseA particle released from rest at the origin has the proper-time trajectoryEliminating proper time gives the semicubical parabolaThis signed coefficient assumes that the positive $x$ direction is chosen along $\mathbf{E}$; reversing that axis changes the signs of both $x^3$ and its coefficient.

A Banach space is a normed vector space over the real numbers or complex numbers that is complete for the metric $d(x,y)=\Vert x-y\Vert$. Explicitly, whenever a sequence $(x_n)$ satisfiesthere is an element $x$ of the same space with $\Vert x_n-x\Vert \to 0$. The hypothesis says that the approximations eventually agree to any requested accuracy; completeness says that they describe an actual object in the space. The Cauchy sequence condition can be checked before anyone knows what that object is.

This makes a Banach space a natural setting for constructing solutions by successive improvements. If corrections $v_k$ have ${\sum }_k\Vert v_k\Vert <\infty$, their partial sums form a Cauchy sequence by the triangle inequality, and completeness supplies their sum. In fact, convergence of every such absolutely summable vector series characterizes a Banach space. The contraction mapping theorem uses precisely this mechanism: each correction is smaller than the preceding one by a fixed factor.^[ref]

Every normed vector space of finite dimension is a Banach space. Infinite-dimensional examples include the absolutely summable sequence space and the l-infinity sequence space; the choice of norm remains part of the specification. A collection of increasingly accurate finite descriptions can require an infinite description for its limit, as the finitely supported sequence example below demonstrates. Completion of a normed space adds these missing limits while preserving all original distances.

A surjective isometry $u:V\to W$ between normed vector spaces preserves every distance:

The supremum norm of a bounded real- or complex-valued function on a set $X$ isFor a continuous function on a compact space, the extreme value theorem makes this supremum a maximum.

The Euclidean norm of $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ is

The trace norm of a finite-dimensional operator $A$ is the sum of its singular values:For a Hermitian operator $A=A_{+}-A_{-}$, it equals $\mathrm{Tr}{A}_{+}+\mathrm{Tr}{A}_{-}$.

For $x\in {\mathbb{R}}^d$, the L1 norm is

This applies normalization of an algebraic curve to the two local branches of a node.

Normalization separates a node's two branches; a rational nodal cubic has normalization ${\mathbb{P}}^1$ and two points above its node.

The geometric genus of an irreducible curve is the genus of its smooth projective normalization.

A unit normal is a normal vector of Euclidean norm one. Given a nonzero normal vector $n_0$, normalization gives $n=n_0/\Vert n_0\Vert$. This makes a dot product $w\cdot n$ measure the signed length of the normal component, without an extra factor from the length of $n_0$. On a hypersurface, the two opposite choices $n$ and $-n$ correspond to opposite normal orientations; a continuous choice on an orientable surface fixes its orientation of a surface.

A closed subset $A$ of a normal space is a countable intersection of open sets exactly when it is the zero set of some continuous function $f:X\to [0,1]$.

Under a surjective homomorphism, the image of a normal subgroup is normal.

The preimage of a normal subgroup under any group homomorphism is normal.

A weak shock has a small relative pressure jump. Its density jump agrees with the reversible adiabatic prediction through second order in the pressure jump, while entropy production first appears at third order.

In the rest frame of a one-dimensional shock, states with velocities $w_0,w_1$ satisfyand

These are the Rankine-Hugoniot conditions specialized to the mass, momentum and energy fluxes of a perfect gas.