For every square matrix over a commutative ring, $\det (A^{\mathsf{T}})=\det (A)$

Depends on

• For $n\ge1$, the determinant over a commutative ring by the Leibniz formula, and $|\det A|$ for a real matrix
• Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose
• The sign is a homomorphism $S_n\to\{+1,-1\}$, surjective exactly when $n\ge 2$
• Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule

Used by

• Orthogonal and unitary operators form groups, and their determinants have modulus one Corollary
• Polar integration may discard the cut locus Corollary
• The determinant is alternating and multilinear in the rows as well as in the columns Corollary
• Full-rank lattices, covolume, and the dual lattice Definition
• A metabolic Seifert matrix has Fox–Milnor factorization Lemma
• Changing an error-contrast basis changes the REML likelihood by a constant Lemma
• Chart and partition independence of surface measure Lemma
• Cyclotomic Schur weight Gram determinants are products of monic semistandard norms Lemma
• For a finite Galois extension, (αⱼ) is a base-field basis exactly when the matrix (σᵢαⱼ) is invertible Lemma
• Gaussian integral for a positive-definite quadratic form Lemma
• Orthogonality of the lattice characters over a fundamental domain Lemma
• Sphere graph charts, surface density, and a finite partition Lemma
• Strict concavity of log determinant on positive-definite matrices Lemma
• The adjugate gives an explicit inverse formula over a field Lemma
• The determinant quotient det g_Ω/K_Ω is a biholomorphic invariant Lemma
• The elliptical-potential bound for ridge design matrices Lemma
• The local intersection sign of graph against diagonal is sign det(I-Df) Lemma
• The quaternion double cover generates the third homotopy group of SO(3) Lemma
• The ridge information matrix is positive definite and invertible Lemma
• Trace control of the regularized design determinant Lemma
• A Gram determinant is nonnegative and is positive exactly when the vector list is linearly independent Theorem
• A positive-definite multivariate normal law has its Gaussian density Theorem
• A proved rational shift bound implies low-rank ADI convergence Theorem
• Exact Two-Qubit Cartan Decomposition Theorem
• Laplace expansion computes the determinant along every row and every column over a commutative ring Theorem
• One-Qubit Euler Decomposition Theorem
• Positive-definite multivariate Gaussian density is normalized Theorem
• Sylvester's criterion: a real symmetric n× n matrix with n≥1 is positive definite if and only if all leading principal minors are positive Theorem
• The Gram formula gives a well-defined positive-definite inner product on exterior powers, and ‖v₁∧⋯∧ vₖ‖² is the Gram determinant Theorem

Dependency tree · two levels

22 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, Theorem 4.23 (standard reference, not scraped)
• P. Massot, Structures algébriques fondamentales, Proposition 6.4.3 (standard reference, not scraped)