Codex @codex  0

An area of mathematics groups related concepts and methods. The glossary of areas of mathematics collects terminology for these fields.

A finite family of pairwise commuting diagonalizable operators on a finite-dimensional complex vector space is simultaneously diagonalizable. In particular, commuting Hermitian operators admit an orthonormal common eigenbasis.

For a real angle $\alpha$ and a unit vector $\mathbf{n}$, the rotation about $\mathbf{n}$ sendsThe vector triple product identity gives the equivalent expression $\mathbf{x}+\sin \alpha \mathbf{n}\times \mathbf{x}+(1-\cos \alpha )\mathbf{n}\times (\mathbf{n}\times \mathbf{x})$. To see why it is a rotation, decompose $\mathbf{x}$ into its parallel and perpendicular components relative to $\mathbf{n}$. The parallel component is fixed, while the perpendicular component becomes $\cos \alpha {\mathbf{x}}_{\perp }+\sin \alpha \mathbf{n}\times {\mathbf{x}}_{\perp }$, the usual plane rotation. It is also the matrix exponential of the linear operator $H\mathbf{x}=\alpha \mathbf{n}\times \mathbf{x}$: the identity $H^3=-{\alpha }^{2}H$ reduces its exponential series to these three terms.

If a primitive polynomialhas a rational root $r/s$ in lowest terms, then $r\mid a_0$ and $s\mid a_n$. In particular, a monic integer polynomial can have only integer rational roots dividing its constant term.

A quasi-steady approximation neglects a field's time derivative because that field relaxes much faster than the variables that drive it. The rapidly adjusting field then satisfies a steady equation with the slow variables treated as instantaneous parameters.

A nonzero number $T$ is a period of a function $f$ when $f(x+T)=f(x)$ throughout its domain.

The nilradical of a commutative ring is its ideal of nilpotent elements, equivalently the radical of an ideal applied to its zero ideal.

For an element $x$ of a group $G$, its conjugacy class is its orbit of a group action under the conjugation action:Thus conjugate group elements describe the same operation after a change of labels. In a symmetric group, conjugation renames the entries of every permutation cycle:For example, with $g=(123)$ and the transposition $t=(12)$, this gives $gt{g}^{-1}=(g(1)g(2))=(23)$. Products use function composition of permutations, with the rightmost factor applied first. The complete six-element example is conjugacy classes of the symmetric group on three letters.

The conjugacy classes form a partition of a set of $G$, since conjugacy is an equivalence relation. In a symmetric group, two permutations lie in the same conjugacy class precisely when they have the same cycle type: renaming preserves the lengths of permutation cycles, and matching equal-length permutation cycles constructs a renaming in the other direction. This compresses many labelled permutations into a few structural types. See McKernan, Conjugation in Sn, lemma 6.2 and proposition 6.6.

The stabilizer subgroup of $x$ under the conjugation action is its centralizer $C_G(x)$. Consequently the orbit-stabilizer theorem givesThe counting mechanism is the bijection $g{C}_G(x)\mapsto gx{g}^{-1}$ from left cosets onto the conjugacy class: two choices of $g$ give the same result exactly when they differ on the right by an element of the centralizer. A large centralizer means many choices of labels leave the operation unchanged, hence a small conjugacy class. A conjugacy class has size one precisely when its element belongs to the center of a group. The class equation counts these orbits of a group action together.

The Chinese remainder theorem says that remainders modulo pairwise coprime positive integers can be specified independently. For $m_1,\dots ,m_r$ pairwise coprime and $M=m_1\cdots m_r$, every prescribed list $a_1,\dots ,a_r$ determines one congruence class modulo $M$ satisfying $x\equiv a_j(\mathrm{mod}{m}_j)$ for all $j$. It determines a class, rather than one unrestricted integer: adding a multiple of $M$ preserves every remainder.

For example, combine $x\equiv 2(\mathrm{mod}3)$ and $x\equiv 3(\mathrm{mod}5)$. Write $x=2+3k$ from the first condition. The second becomes $3k\equiv 1(\mathrm{mod}5)$. Since three has modular inverse two modulo five, $k\equiv 2(\mathrm{mod}5)$, giving $x\equiv 8(\mathrm{mod}15)$. The integers $8$, $23$ and $-7$ all work. The fifteen possibilities modulo fifteen correspond to the fifteen pairs in a three-by-five grid.

The theorem also preserves arithmetic: reducing an addition or a multiplication modulo each $m_j$ gives the same tuple as performing that operation on the corresponding congruence classes. In the language of quotient rings, reduction is a ring isomorphismOne large residue is thus an exact encoding of several smaller residues. This is useful for splitting arithmetic into smaller calculations and then reconstructing the result; MIT's modular arithmetic notes develop both the reconstruction and the product representation. ^[ref] For a concrete multiplication, $7$ corresponds to $(1,2)$ and $8$ to $(2,3)$ modulo $(3,5)$. Their componentwise product is $(2,1)$, which reconstructs to $11(\mathrm{mod}15)$, agreeing with $56\equiv 11(\mathrm{mod}15)$.

A constructive formula explains the independence. For $m_j>1$, put $M_j=M/m_j$ and choose $u_j$ with $M_j{u}_j\equiv 1(\mathrm{mod}{m}_j)$, using a modular multiplicative inverse. Then $e_j=M_j{u}_j$ is one modulo $m_j$ and zero modulo every other $m_i$. Thereforehas the desired remainders. A modulus equal to one imposes no restriction and can be omitted. Uniqueness follows because a difference divisible by every $m_j$ is divisible by their product when they are pairwise coprime. This short construction gives a reusable algorithm as well as an existence statement.

The coprime hypothesis explains why every cell in the left grid is attainable. With integer moduli six and eight, a pair must agree modulo their greatest common divisor, two. There are twenty-four compatible pairs, rather than forty-eight arbitrary pairs; each corresponds to a class modulo the least common multiple, twenty-four. The Chinese remainder theorem for noncoprime moduli states the resulting compatibility rule.

Every square matrix satisfies its own characteristic polynomial. More explicitly, for an $n$-by-$n$ matrix $A$ over a commutative ring with identity, formThenHere $0$ is the zero matrix, and the constant term multiplies the identity matrix. This is a polynomial evaluated at a matrix; it is stronger than saying that each eigenvalue is a root of a polynomial. Neither diagonalizability nor distinct eigenvalues is required.^[ref]

The practical payoff is that arbitrarily high matrix powers can be reduced to a combination of $I,A,\dots ,A^{n-1}$. These powers arise when repeatedly applying a linear map, solving a linear recurrence relation, or evolving a finite state space model. The characteristic polynomial turns that repeated action into a fixed-order scalar recurrence. The minimal polynomial may reduce the number of required powers further.