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.

A matrix pencil is an affine one-parameter family of matrices. For square matrices of size $m$, its determinant is a polynomial of degree at most $m$. If $B$ is invertible, its leading coefficient is $\det B\ne 0$, so at most $m$ distinct parameters give a singular matrix. This translates linear dependence in a family of bases into polynomial root counting.

A matrix norm is a norm on a vector space of matrices; common choices include induced operator norms and the Frobenius norm. A submultiplicative matrix norm additionally satisfies $\Vert AB\Vert \le \Vert A\Vert \Vert B\Vert$. The matrix 2-norm is the norm induced by Euclidean norms and equals the largest singular value.

A Cauchy matrix has entries $1/(x_i+y_j)$ with nonzero denominators. If $x_i=y_i>0$ are distinct, it is positive definite: $C_{ij}={\int }_0^{\infty }{e}^{-x_{i}s}{e}^{-x_{j}s}ds$ is the Gram matrix of linearly independent exponential functions. This form occurs in finite-rank inverse scattering transform reconstructions.

An integer matrix is a matrix whose entries are integers. Sums and products of compatible integer matrices remain integer matrices, but a nonsingular square integer matrix can have a nonintegral inverse matrix. The unimodular matrix condition characterizes exactly when that inverse matrix also has integer entries.

In a singular first-order equation with coefficient $x+ϵ$, an outer correction proportional to $ϵ\log x$ cannot be evaluated at the endpoint. The inner scale $x=ϵX$ changes it to $ϵ\log ϵ+ϵ\log (1+X)$. Matching retains the logarithmic constant, which is larger than a plain order-epsilon correction. For $(x+ϵ)y^{\prime }=ϵy$, $y(1)=1$, the exact solution $[(x+ϵ)/(1+ϵ){]}^{ϵ}$ verifies the displayed endpoint asymptotic and its remainder.

A decaying algebraic tail in an inner asymptotic expansion can become a fractional-order outer correction after rescaling. For an inner tail proportional to $r^{-1/2}$ and outer variable $x=ϵr$, the outer correction is proportional to ${ϵ}^{1/2}{x}^{-1/2}$. If the next outer correction contains $ϵ\log x$, expressing it in the inner variable produces both $ϵ\log r$ and $ϵ\log ϵ$. Omitting the latter loses a term larger than a plain $O(ϵ)$ correction. Matching must include all these scales and constants, rather than only the leading algebraic powers.

Multiplication by $x^2$ makes this example exactly integrable once: $ϵx^{2}y\prime +(x^2-2{ϵ}^3)y=C$. Its homogeneous solution is $\exp (-x/ϵ-2{ϵ}^2/x)$. A solution with $y(1)={ϵ}^3$ that remains uniformly bounded needs $C\sim {ϵ}^3$, hence $y(0)\sim -1/2$. The scales $x=O(ϵ)$ and $x=O({ϵ}^2)$ connect the small outer solution to the order-one endpoint value. Bounded homogeneous contributions remain invisible to algebraic matching at the right boundary.

For fixed $k>0$ and endpoint values $y(0)=0$, $y(1)=1$, the outer solution is $y\sim {\epsilon }^2{k}^2$. The left boundary layer has width $\epsilon$. At the right, a layer of width $\sqrt{\epsilon }$ carries the order-one rise, and a translated layer of width $\epsilon$ connects its inner edge to the small plateau. The dominant balance depends on both amplitude and derivative scale; using the same scale at both ends loses a necessary match.

An asymptotic matching condition requires the inner expansion and outer expansion, re-expressed in a common overlap region, to agree to the retained order. Both function values and slopes may constrain the free constants. In the square-root reaction problem with nested boundary layers, the shared small plateau sets the first integral of a square-root reaction layer, while its large-field fourth-power tail fixes the neighboring broad layer.

For $q>0$, matching $Y\to -q$, $Y^{\prime }\to 0$ fixes the first integral to $(Y^{\prime }{)}^2=(q-2Y)(Y+q{)}^2/3$. Hence endpoint values above $q/2$ cannot match this outer branch. The upper family is $Y=-q+(3q/2){\mathrm{sech}}^2[\sqrt{q}(\xi -{\xi }_0)/2]$. An endpoint value strictly between $-q$ and $q/2$ admits two translations, one initially approaching $-q$ and one initially moving towards the turning value $q/2$. Values below $-q$ give the unique lower family $Y=-q-(3q/2){\mathrm{csch}}^2[\sqrt{q}(\xi +{\xi }_0)/2]$ with ${\xi }_0>0$. Matching to zero instead gives $(Y^{\prime }{)}^2=-q{Y}^2-2Y^3/3$, which forbids a nonconstant real approach to zero. These are leading boundary layer classifications, not exact multiplicity theorems at degenerate endpoint data.

A growing linear mode can make a formally small derivative nonlinearity order one at large distance. After the crossover scaling, the equation $X{H}^{\prime \prime }-H^{\prime }+(H^{\prime }{)}^2=0$ reduces to a logistic differential equation in $\log X$ for $H^{\prime }$. Matching $H\sim X^2/2$ at small $X$ gives $H^{\prime }=X/(1+X)$ and the displayed primitive with $H(0)=0$. It changes from quadratic growth to linear growth without a finite turning singularity.

For $ϵy^{\prime \prime }-(1+x)y^{\prime }+y=0$, $y(0)=1$, $y(1)=1+ϵ$, the exact outer solution is $1+x$ and the mismatch is at the right endpoint. In $X=(1-x)/ϵ$, matching gives $Y_0=2-e^{-2X}$ and $Y_1=-X+(1-X-X^2/2)e^{-2X}$. Differentiation at $X=0$ gives the displayed endpoint slope. The order-one term requires the first inner correction; differentiating only the leading boundary layer loses it.

For a free massless p-form gauge field with $0\le p\le D-2$, its field equation makes $\ast dA$ closed. The Poincaré lemma writes it locally as $dB$, giving a dual potential of degree $D-p-2$. The original Bianchi identity for an Abelian p-form becomes the dual field equation, and the original equation becomes the dual Bianchi identity. The two descriptions have equal little group counts $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)$. Global fluxes and top forms require separate treatment.

// Target: quantum-field-theory.bigb

For a free massive p-form field, the displayed dual has degree $D-p-1$. The massive field equation reconstructs $A$ locally from $m^{-1}\ast dB$ with the signature-dependent sign. It also gives $\delta B=0$ and $\delta dB+m^{2}B=0$. The polarization counts agree because $\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{D-p-1}\right)$. The construction depends on nonzero mass and therefore differs from massless p-form duality by one potential rank.

// Target: physics.bigb

For $p\ge 1$, introduce a $(p-1)$-form $C$ and replace the mass term of a massive p-form field by one built from $A-m^{-1}dC$. The transformations $A\mapsto A+d\Lambda$, $C\mapsto C+m\Lambda$ leave this combination and $dA$ invariant. Gauge fixing $C=0$ recovers the original massive theory. Its massless-component counts add as $\left(\genfrac{}{}{0ex}{}{D-2}{p}\right)+\left(\genfrac{}{}{0ex}{}{D-2}{p-1}\right)=\left(\genfrac{}{}{0ex}{}{D-1}{p}\right)$, explaining the longitudinal degrees of freedom and the corresponding assembly of nonzero Kaluza-Klein modes.