The determinant of a triangular matrix is the product of its diagonal entries

Depends on

• Upper triangular, lower triangular and diagonal square matrices over a commutative ring
• For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
• Inversions, inversion number, the sign $\operatorname{sgn}(\sigma)=(-1)^{\operatorname{inv}(\sigma)}$, and even and odd permutations

Used by

• Finite-candidate information design Algorithm
• In positive dimension the determinant of an operator is computed from a representing matrix by row reduction, tracking swaps and row scalings Corollary
• The index of a full-rank subgroup of ℤⁿ is the absolute determinant of a generating matrix Corollary
• The critical Riesz potential can diverge and be essentially unbounded Counterexample
• Standard subgroups of finite general linear groups Definition
• A sparse 4×4 determinant is 30 by both cofactor expansion and row reduction Example
• A unit-scale affine coupling has a triangular Jacobian Example
• Brownian finite-dimensional density Example
• Chebyshev extremals and the exact disk Fekete polynomial Example
• Dilation determines the Riesz-potential target exponent Example
• Regular and singular torus characters in GL₃(F_q) Example
• Row operations track determinant correctly for a singular triangular matrix over ℤ/6 Example
• The Coxeter complex of I₂(5): the circle triangulated by the ten chambers Example
• The Lebesgue measure of the image of the unit cube under an explicit linear map of the plane and of three-space Example
• A coordinate scaling and a coordinate transposition send the unit cube to a set of measure equal to the absolute value of the determinant Lemma
• A determinant and Frobenius norm give a certified lower bound for small-block separation Lemma
• A Hermitian two-by-two pivot inertia is determined by determinant and trace signs Lemma
• A nonsingular principal minor of the symmetrized Cartan matrix of size the rank Lemma
• A shear sends the unit cube to a set of Lebesgue measure one Lemma
• A uniform positive eigenvalue floor controls log-determinant error Lemma
• An explicit root-separation bound for squarefree integer polynomials Lemma
• Bfgs trace and determinant identities give a matrix potential inequality Lemma
• Bounded compact data give an everywhere finite Newtonian potential Lemma
• Determinants and kernel quotients of the model Bergman metrics Lemma
• Fixed-memory L-Bfgs curvature and scale bounds imply uniform spectral bounds Lemma
• For symmetric M=beginpmatrixA&BB^T&C endpmatrix with A invertible, a block-unitriangular congruence gives A⊕(C-B^TA⁻¹B) and factors det M Lemma
• Gaussian integral for a positive-definite quadratic form Lemma
• Holomorphic Morse normalization and Gaussian contour Lemma
• Length-additive products of the standard intertwiners Lemma
• Near and far bounds for a Riesz potential Lemma
• Positively oriented bases of an oriented vector space are path-connected Lemma
• Powers of a Schur-stable matrix decay geometrically up to a polynomial factor Lemma
• Strict concavity of log determinant on positive-definite matrices Lemma
• The conditional Gaussian mixture is a nonnegative supermartingale Lemma
• The elliptical-potential bound for ridge design matrices Lemma
• The equal-coordinate rank-one principal series of GL₂ Lemma
• The log determinant is concave on positive-definite matrices Lemma
• The rank-one Hecke parameter for equal torus characters Lemma
• The real Jacobian determinant of a complex-linear automorphism is the squared modulus of its complex determinant Lemma
• The ridge information matrix is positive definite and invertible Lemma

…and 26 more results.

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• S. New, MATH 146 Linear Algebra 1 Lecture Notes, Ch. 4 (standard reference, not scraped)
• D. Margalit and J. Rabinoff, Interactive Linear Algebra, §4.1 (standard reference, not scraped)