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If the canonical representative of a subharmonic distribution on a connected domain attains a finite global maximum at an interior point, it is constant. The ball mean inequality forces equality almost everywhere on a small ball around the maximum. The canonical representative then equals the maximum everywhere on that ball. The maximum set is open by this argument and closed by upper semicontinuity; connectedness makes it the whole domain.

For independent $U[0,\theta ]$ observations, the likelihood decreases as ${\theta }^{-n}$ on $[{\max }_i{Y}_i,\infty )$, so its maximizer is the sample maximum almost surely. Its exact distribution is ${\mathbb{P}}_{\theta }({\hat{\theta }}_n\le y)=(y/\theta {)}^n$ for $0\le y\le \theta$. This proves consistency and an endpoint error of order $1/n$, with the displayed exponential limit. Parameter-dependent support violates the usual regularity condition behind nondegenerate square-root-sample-size normal limits; the ordinary interior score is not centered.

A likelihood need not attain its supremum when the parameter space excludes a boundary point. For example, with parameter $0<\theta \le 1$ and a zero Poisson count, the likelihood $e^{-\theta }$ is strictly decreasing and has no maximum. Extending the parameter space to include zero gives an extended maximum-likelihood estimate there. A proposed estimator formula involving that boundary value must distinguish the extended estimate from an actual maximizer in the original space.

For a compact torus $T\cong U(1{)}^r$, its continuous one-dimensional characters form a free abelian group of rank $r$. They are precisely $z\mapsto z_1^{m_1}\cdots z_r^{m_r}$ with integer exponents. This follows from the classification of representations of the circle group, applied to each factor. These characters are the integral group weights occurring upon restricting a group representation to a maximal torus; global integrality must be distinguished from integrality against coroots for a simply connected semisimple covering group.

Give the Lie algebra of a compact Lie group an inner product invariant under the Adjoint representation. For a maximal torus $T$ with Lie algebra $\mathfrak{t}$, its Lie-algebra centralizer is $\mathfrak{t}$: any additional commuting vector would generate, together with $T$, a larger compact connected abelian subgroup. The torus weight decomposition therefore supplies $H\in \mathfrak{t}$ whose centralizer is exactly $\mathfrak{t}$, by avoiding finitely many nonzero-weight hyperplanes. For any $X$, maximize $⟨{\mathrm{Ad}}_{g}X,H⟩$ on the group. Its directional derivative gives $⟨Z,[{\mathrm{Ad}}_{g}X,H]⟩=0$ for every $Z$, so ${\mathrm{Ad}}_{g}X\in \mathfrak{t}$. For a connected compact group, the bi-invariant Riemannian metric and Hopf-Rinow theorem make the exponential onto; exponentiating this conjugacy puts every element in a conjugate of $T$.

Every involution of $G{L}_3(2)$ is a transvection $I+vf$, because its nilpotent part has square zero and rank one. Two distinct commuting transvections either have common image line or common fixed plane. This gives two types of elementary abelian groups of order four, displayed above. No larger elementary abelian $2$-subgroup is possible: a transvection commuting with all three elements of a common-image group must have that same image, and the dual argument applies to a common-fixed-plane group. Their normalizers are respectively point and plane stabilizers of order $24$. The common fixed-space dimensions one and two distinguish their conjugacy classes.

For a maximal subgroup $M$ of a finite nonabelian simple group, take a minimal nontrivial normal subgroup $N$ of $M$. It is a characteristically simple group. Its normalizer contains $M$ but cannot be the whole simple group, since $1<N\le M<G$. Maximality gives equality.

Two maximal subgroups of a finite soluble group either multiply to the whole group or are conjugate. Reduce modulo a nontrivial normal subgroup contained in either maximum. If no such subgroup exists, both maxima complement a minimal normal elementary abelian subgroup $K$. Their actions on $K$ are faithful and irreducible. A minimal normal subgroup of $G/K$ has characteristic different from that of $K$, since a normal p-subgroup has nonzero fixed points on a characteristic-p module. The intersections of the two maxima with its preimage are conjugate Sylow subgroups. Each maximum is the normalizer of its nontrivial intersection, proving conjugacy.

Up to conjugation, the proper maximal subgroups of $S_n$ containing a three-cycle are $A_n$; the unequal-part subset stabilizers $S_k\times S_{n-k}$ for $1\le k<n/2$ with $n-k\ge 3$; and the full block stabilizers $S_a\wr S_b$ for $n=ab$, $a\ge 3$, $b\ge 2$. The primitive three-cycle criterion treats the primitive case. A nontrivial block system gives the full permutation wreath product in the imprimitive case. These block stabilizers are maximal because any extra permutation produces a transposition connecting two different blocks; conjugating by the block stabilizer then gives every cross-block transposition, hence the symmetric group. Unequal-part subset stabilizers are maximal by the same connected-transposition argument. Equal-part subset stabilizers are not maximal: they lie in the larger stabilizer that can interchange the two parts.

In a complex unital Banach algebra, right multiplication by $a$ induces a bounded module endomorphism $T$ of $A/L$ when $La\subseteq L$. Every nonzero module endomorphism of this simple module is algebraically bijective, since its kernel and range are submodules. Its inverse is bounded by the bounded inverse theorem. Choosing $\lambda$ in the nonempty spectrum of an element of $T$ forces $T-\lambda I=0$. Applying this identity to $1+L$ gives $a-\lambda 1\in L$, with uniqueness because $1\notin L$.

A maximal algebraic torus $T$ is rational for $F$ when $F(T)=T$. Rationality does not require every point of $T$ to be fixed: the finite group $T^F$ is its rational-point group. Similarly a rational Borel subgroup or parabolic subgroup means an $F$-stable algebraic subgroup.

A matrix summability method is regular if it preserves the ordinary limit of every convergent sequence. The Silverman-Toeplitz theorem characterizes regularity by uniformly bounded absolute row sums, entrywise convergence to zero down every fixed column, and convergence of the row sums to one.

An upper Hessenberg matrix has zero entries below the first subdiagonal; the lower version has zero entries above the first superdiagonal. The upper Hessenberg matrix is one of these two forms.

A bidiagonal matrix has nonzero entries only on its main diagonal and one adjacent diagonal. The upper bidiagonal matrix uses the superdiagonal; the lower version uses the subdiagonal.

The matrix transpose interchanges rows and columns: $(A^T{)}_{ij}=A_{ji}$. On a fixed matrix space, $A\mapsto A^T$ is the transposition map, a linear map that preserves the trace norm but is not completely positive in dimensions greater than one.

A matrix decomposition expresses a matrix as a matrix product of factors with simpler structure. An LU decomposition uses triangular factors, a QR decomposition uses an orthogonal matrix and a triangular factor, and a singular value decomposition separates input directions, stretching factors and output directions. The factorization is an identity; the chosen structure determines what it helps compute.

For an original two-by-two example,The outer factors are a lower triangular matrix and an upper triangular matrix, each with unit diagonal as in a unit triangular matrix; the middle factor is a diagonal matrix. Their matrix product is $M$, and the final diagonal entry $5/2$ is the Schur complement of the first entry $2$. This is the scalar version of the factorization under block elimination.

To solve $Mz=(3,4{)}^T$, solve with the three factors in their displayed order from left to right: first the lower triangular factor, then the diagonal factor, then the upper triangular factor. The intermediate vectors are $(3,5/2{)}^T$ and $(3/2,1{)}^T$, giving the concise answer $z=(1,1{)}^T$. A new right-hand side uses the same factors. This is why Reusing LU factors for multiple right-hand sides is useful: factorization records work that otherwise would be repeated, while forward substitution and back substitution apply it cheaply.^[ref]

Different decompositions reveal different features. A Cholesky decomposition uses positive definiteness of a symmetric matrix, while a singular value decomposition also applies to rectangular matrices and exposes matrix rank. The existence and numerical behaviour of the factors depend on their own hypotheses; for example, an unpivoted LU decomposition can fail when a chosen pivot element is zero even though the matrix is an invertible matrix.

The row space of a matrix is the span of its rows, and its column space is the span of its columns. Both have dimension equal to the matrix rank, though they lie in different ambient coordinate spaces for a rectangular matrix.

A Hadamard matrix is a real square matrix with entries $+1$ and $-1$ whose rows are pairwise orthogonal. Each row has squared norm $n$, so $H{H}^T=nI$ and $H/\sqrt{n}$ is an orthogonal matrix. In particular $H$ is invertible, with $H^{-1}=H^T/n$. The four-by-four exampleis symmetric and satisfies $H^2=4I$. It supplies an exact intertwiner in four-tile mixed-boundary transplantation between a disk and a nonorientable surface.

An elementary column operation interchanges two columns of a matrix, scales one by a nonzero scalar, or adds a multiple of another column. It is an elementary row operation on the transpose. Hence it has the same three effects on the determinant: changing sign, multiplying by the scale factor, or leaving the determinant unchanged.

A real rectangular matrix is totally nonnegative if every square minor, with row and column indices in increasing order, is nonnegative. A nonnegative bidiagonal matrix is totally nonnegative, and products of compatible totally nonnegative matrices have the same property by the Cauchy–Binet formula. This controls signs of inverse entries and underlies stability and shape-preserving properties of B-spline collocation. Total nonnegativity permits zero minors; strict total positivity requires every such minor to be positive.