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Multiplication by a generator of ${\mathbb{F}}_{q^n}^{\times }$ on the ${\mathbb{F}}_q$-space ${\mathbb{F}}_{q^n}$ gives a cyclic subgroup regular on nonzero vectors. Its normalizer contains the field Frobenius automorphism. In degree three over ${\mathbb{F}}_2$ the full normalizer has order 21.

The maximal subgroups are point stabilizers and plane stabilizers of order 24, and Singer cycle normalizers of order 21. A proper subgroup containing a 7-element has a unique Sylow 7-subgroup, since the other count gives an impossible index-three subgroup of the simple group. A subgroup of order dividing 24 has a fixed point or a three-point orbit on the seven nonzero vectors. A collinear orbit gives an invariant plane; an independent triple gives its fixed vector sum.

Inverse transpose is an automorphism of order at most two. On projective geometry it implements point-hyperplane duality. For $G{L}_3(2)$ it swaps distinct point- and plane-stabilizer classes, so it is outer.

This group preserves a nondegenerate alternating bilinear form on ${\mathbb{F}}_q^{2m}$. Choosing a nonzero vector and a partner of pairing one, then recursing on their orthogonal complement, gives $|S{p}_{2m}(q)|=q^{m^2}{\prod }_{i=1}^m(q^{2i}-1)$. Its center consists of scalars with ${\lambda }^2=1$.

For column vectors with the preserved space first, $A,D$ are invertible and $B$ arbitrary. Thus $P_k\cong {\mathbb{F}}_q^{k(n-k)}⋊(G{L}_k(q)\times G{L}_{n-k}(q))$. Its orbits on $k$-spaces are classified by intersection dimension with the preserved space.

The projective quotient of determinant-one matrices preserving a nondegenerate Hermitian form over ${\mathbb{F}}_{q^2}$, with conjugation $x\mapsto x^q$. In dimension three its natural two-transitive action has $q^3+1$ isotropic points. For $q\ge 3$ its order is $q^3(q^3+1)(q^2-1)/\gcd (3,q+1)$.

The group of invertible linear maps preserving a nondegenerate Hermitian form over a quadratic finite field is, in an orthonormal basis, the displayed matrix group. Here $q^2$ denotes the size of the field on which the matrices act; some authors denote this same group by $U_n(q)$. Imposing determinant one gives the special unitary subgroup, and subsequently quotienting its scalar center gives the projective special unitary group over a finite field.

If $k_n(q)$ counts conjugacy classes in $G{L}_n(q)$, then ${\sum }_{n\ge 0}{k}_n(q)t^n={\prod }_{i\ge 1}(1-t^i)/(1-q{t}^i)$. Associate a partition to each irreducible polynomial other than t. The Euler product over those polynomials counts monic polynomials with nonzero constant term and is $(1-u)/(1-qu)$.

Assume $a$ and $a^b$ algebraic with algebraic irrational $b$. Apply Siegel lemma to construct a small-coefficient polynomial $P$ vanishing at the algebraic pairs $(s+tb,a^s(a^b{)}^t)$ on a large square grid. The function $P(z,e^{\lambda z})$ is a nonzero exponential polynomial. Its many initial zeros make later grid values exponentially tiny, whereas an algebraic norm estimate forbids such tiny nonzero values. Iterating extends the zero grid indefinitely. Its quadratic zero count in disks contradicts the linear zero-count growth of an exponential polynomial.

In the group algebra $\mathbb{C}{S}_n$, setThese elements commute: for each triple $a<i<j$, the only overlapping contributions cancel as $[(ai),(ij)+(aj)]=0$. They generate the Gelfand–Tsetlin algebra, and on a standard Young tableau vector $X_r$ acts by the Content of a Young-diagram cell containing $r$.

For a chaotic incompressible planar flow with Lyapunov exponent pair $\lambda ,-\lambda$, an initially broad Gaussian scalar packet in a linear incompressible flow stretches and compresses exponentially while diffusion is negligible. Its compressed variance stops decreasing around $\kappa /\lambda$, the Batchelor scalar microscale balance, while the larger variance continues to grow. In a constant principal strain, the normalized covariance eigenvalues obey ${\dot{j}}_{\pm }=\pm 2\lambda j_{\pm }+2\kappa$, proving that the compressed one approaches $\kappa /\lambda$. Random finite-correlation strain instead gives a fluctuating compressed width; stationarity and integrability assumptions are required to turn this into exponential statistics of the packet amplitude.

A centered continuous stationary scalar Gaussian process of positive variance can be Markov only if its covariance satisfies the displayed equation for $s,t\ge 0$. Gaussian conditioning makes the future conditional mean $C(t)X_s/C(0)$; the Markov property then forces the equation by conditioning a future observation on the present and past. Conversely this equation makes the Gaussian residual future independent of the past, proving the Markov property. In the nondegenerate decaying case the covariance is a single exponential. A sum $c_1{e}^{-a_1|t|}+c_2{e}^{-a_2|t|}$ with positive coefficients and distinct positive decay rates fails the equation and hence is not scalar Markov.

Let $X_0=0$ and let $X$ be a centered Gaussian process whose increments are independent of the past filtration, with deterministic variance $v(t)$. Its displayed exponential is a positive martingale. Indeed, $X_t-X_s$ is a centered Gaussian random variable with variance $v(t)-v(s)$, so the Gaussian moment-generating function givesThe sign-reversed process has the same variance and the same conclusion. Continuity of paths is unnecessary for this conditional-expectation argument.

Gaussian process regression estimates an unknown function by assigning its values a Gaussian process law, then conditioning on observed values, often with independent Gaussian noise. The covariance function expresses the proposed relation between nearby inputs; the conditional multivariate normal distribution gives the resulting Gaussian process regression posterior. Its conditional mean provides a prediction and its conditional variance quantifies uncertainty under the chosen model. The index set can represent spatial locations or other inputs as well as time.

For $H=p^2/(2m)+V(q)$, time slices of length $\delta$ use $N$ momentum measures $d{p}_j/(2\pi \hslash )$ and $N-1$ intermediate coordinate measures. Each momentum Gaussian integral gives $[m/(2\pi i\hslash \delta ){]}^{1/2}\exp [im(q_j-q_{j-1}{)}^2/(2\hslash \delta )]$. The resulting configuration integral has normalization $[m/(2\pi i\hslash \delta ){]}^{N/2}$. A common endpoint convention for $V$ and the Trotter product formula identify both limits with the same operator transition kernel; a damping or imaginary-time prescription defines the oscillatory integrals.

// Target: perturbative-quantum-field-theory.bigb

An elementary row operation interchanges two rows of a matrix, multiplies one row by a nonzero scalar, or adds a scalar multiple of another row to it. These invertible operations are used in Gaussian elimination. For a square matrix, interchange changes the sign of its determinant, multiplication scales its determinant by the same scalar, and adding another row leaves its determinant unchanged, by multilinearity and alternation.

A matrix over a field $\mathbb{F}$ is in row echelon form when all zero rows follow the nonzero rows, each nonzero row's first nonzero entry lies strictly to the right of the preceding row's first nonzero entry, and all entries below each such leading entry vanish. These leading entries are its pivot elements. They need not be one, and entries above them need not vanish.^[ref]

The staircase turns a linear system into a sequence of simpler equations. For example,is in row echelon form, with pivot columns 1 and 2. Its second equation determines $x_2$ once $x_3$ is chosen; its first then determines $x_1$ by back substitution. It is not in reduced row echelon form, because the second pivot column contains a nonzero entry above its pivot element. Neither being square nor having a nonzero entry in every row is required.

Elementary row operations bring every matrix to row echelon form: choose a nonzero entry in the leftmost available nonzero column, interchange rows, clear entries below it, and repeat on the remaining rows and later columns. Skip columns with no remaining nonzero entry. Clearing above each pivot element and scaling it to one completes reduced row echelon form.^[ref]

The nonzero rows of an row echelon form are linearly independent: in a vanishing linear combination, inspecting the first pivot column forces the first coefficient to be zero, then the next forces the second, and so on. They give a basis of the original row space. The number of pivot elements is both the row rank and the column rank. To recover a basis of the original column space, select the corresponding original pivot columns; the reduced column vectors generally belong to a different column space.

Here the arithmetic is exact in the specified field. For floating-point arithmetic, deciding whether a computed entry counts as zero requires a numerical criterion; see Tiny-pivot roundoff in Gaussian elimination and condition number. An exact row echelon form classification alone does not settle numerical stability.

At each step of Gaussian elimination, select the largest absolute entry in the uneliminated part of the current column and interchange rows to put it on the diagonal. A nonsingular trailing Schur complement has a nonzero entry in its first column, so this procedure always finds a nonzero pivot in exact arithmetic. The permutation matrix $P$ records row interchanges; earlier multipliers must be interchanged too when assembling the lower-triangular factor. This yields the LU decomposition $PA=LU$ with unit diagonal in $L$. Searching within columns and interchanging rows is distinct from interchanging columns.

Sparse Gaussian elimination solves a linear system by Gaussian elimination while exploiting its sparse matrix structure to avoid storing zeros and performing unnecessary arithmetic operations. The catch is that zeros in the input need not remain zero: eliminating a shared unknown can couple two equations that previously had no direct coupling. These new positions are fill-in, and their number can depend strongly on the order of elimination.

For a real symmetric positive-definite matrix, the usual direct method is a Cholesky decomposition. Choose an ordering, use symbolic factorization to allocate the predicted factor entries, compute their values, then solve two triangular systems. Compressed sparse storage makes the reduced storage useful. General LU decomposition may also need numerical partial pivoting, so an ordering chosen for sparsity can change during factorization.^[ref]

For a fixed ordering, let $P$ be the permutation matrix that puts the unknowns in that order. Permute both rows and columns:Here $C$ is the lower triangular matrix in the Cholesky decomposition of $B$; $z=Px$. Forward substitution and back substitution give the original solution. The simultaneous permutation preserves the symmetric positive-definite matrix property because $v^{T}Bv=(P^{T}v{)}^{T}A(P^{T}v)>0$ for every nonzero vector $v$. A row permutation alone generally loses the symmetric matrix structure.

The local mechanism is a Schur complement. If the active pivot element is $a_{kk}$, then for remaining indices $i,j$,Even when $a_{ij}=0$, two nonzero couplings through $k$ make the second term nonzero. Thus the elimination graph removes $k$ and joins its remaining neighbors into a clique. The fill-path criterion for symmetric elimination extends this local rule across several steps. The elimination tree organizes the resulting dependencies; the minimum degree algorithm and nested dissection choose orderings to control them.

This connection between graphs and linear systems was already explicit in Donald J. Rose, R. Endre Tarjan and George S. Lueker's 1976 study of fill-in and perfect elimination orderings.^[ref]

For $i>k$, an elimination multiplier is the scalar $l_{ik}=a_{ik}^{(k-1)}/a_{kk}^{(k-1)}$ used in the elementary row operation $R_i\leftarrow R_i-l_{ik}{R}_k$. This creates a zero below the pivot element. The remaining entries change by $a_{ij}\leftarrow a_{ij}-l_{ik}{a}_{kj}$ for $j>k$. Storing these elimination multipliers gives the subdiagonal entries of $L$ in LU decomposition; later row interchanges must also reorder the already stored coefficients.