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Orthogonal Householder similarities reduce a real symmetric matrix to symmetric tridiagonal form.

Successive Householder reflections zero subdiagonal column entries and factor a matrix into an orthogonal factor and an upper-triangular factor.

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.

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.