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Consider assemblies of congruent polygonal tiles with same-labelled faces paired by reflection and pure Neumann boundary conditions on unpaired faces. An invertible intertwiner for all face permutation matrices preserves the Euler characteristic. The number of tiles is the dimension of the representation. For each face label the number of edge classes is $(n+\mathrm{tr}{M}_s)/2$. For each vertex label the number of vertex classes is the number of orbits of the subgroup generated by incident-face involutions, hence the dimension of its invariant vectors. Intertwining preserves all these numbers. Summing the vertex and edge counts proves equal $V-E+F$, including boundary corners. This argument uses genuine permutation matrices; signed matrices encoding mixed boundary conditions do not give the same topological count.

A signed permutation matrix has exactly one nonzero entry in each row and column, and every nonzero entry is $1$ or $-1$. It is a product of a diagonal sign matrix and a permutation matrix.

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]

A stoquastic circuit verifier uses reversible classical quantum gates, represented by permutation matrices, with zero and plus-state ancilla qubits and a final Hadamard basis output measurement. A basis input remains an entrywise nonnegative state; a general quantum witness need not be a basis input. The resulting acceptance structure is used in StoqMA.

A stoquastic Hamiltonian is a real Hermitian Hamiltonian operator with nonpositive off-diagonal entries in a specified basis, usually the computational basis. The property depends on that basis. Permutation matrices in a quantum circuit's propagation terms and $|-⟩⟨-|$ input penalties give a useful family of such operators.

This doubly stochastic matrix replaces two values $a,b$ by $ta+(1-t)b$ and $(1-t)a+tb$, preserving their sum. It fixes all other coordinates when embedded in a larger matrix. A finite product of such averaging matrices and permutation matrices is again doubly stochastic.