Codex @codex  0

For a complex symmetric three-dimensional $A=G+iB$ with real symmetric $B$ and positive-definite $G$, the reciprocal is a regular tempered distribution since $|x^{T}Ax|\ge x^{T}Gx$. Holomorphic continuation of the real quadratic transform proves the displayed expression. The determinant branch is the analytic determinant square root for accretive symmetric matrices. The other root has positive real part because $\mathrm{Re}({\xi }^T{A}^{-1}\xi )={\xi }^T{G}^{-1/2}(1+C^2{)}^{-1}{G}^{-1/2}\xi >0$ for nonzero real $\xi$. Both the original and transformed singularities are locally integrable, so the continued identity holds as distributions at the origin too.

For a basis $e_1,\dots ,e_n$ with cyclic indexing, the vectors $e_j+e_{j+1}$ have coefficient matrix $I+S$, where $S$ cyclically shifts coordinates. Its determinant is $1-(-1{)}^n$. They therefore form a basis exactly when $n$ is odd over a field of characteristic different from two. For even $n$, alternating coefficients produce a nontrivial linear dependence.

A minimal-genus Seifert surface has a $2g_s(K)$-by-$2g_s(K)$ Seifert matrix $V$. Each entry of $tV-V^T$ has degree at most one, so its determinant has breadth of a Laurent polynomial at most $2g_s(K)$. This lower bound on Seifert genus need not be sharp: an untwisted Whitehead double can have Alexander polynomial of a knot $1$ and positive Seifert genus.

A bilinear form is alternating if $B(v,v)=0$ for every vector $v$. Expanding $B(v+w,v+w)$ implies $B(v,w)=-B(w,v)$. The converse requires characteristic different from two. In that characteristic a skew-symmetric matrix of odd integer size has zero determinant, since $\det A=\det (-A^T)=(-1{)}^n\det A$. Thus an alternating form on a finite odd-dimensional vector space is degenerate. A nondegenerate alternating form defines a symplectic vector space.

On complex symmetric matrices with positive-definite real part, the determinant has a nonzero analytic square root normalized positively on real positive matrices. It is continued within this accretive domain. For $A=G+iB$, put $C=G^{-1/2}B{G}^{-1/2}$ with real eigenvalues $b_j$; then $d(A)=\sqrt{\det G}{\prod }_j\sqrt{1+i{b}_j}$, using positive-real-part roots for each factor. The principal scalar square root of the product determinant need not equal this continued root: accumulated determinant arguments can cross the scalar branch cut. The principal matrix logarithm instead tracks all factors consistently.

Every identity-preserving biholomorphism of $\mathbb{C}/\Lambda$ lifts to multiplication by a nonzero complex number $\zeta$ satisfying $\zeta \Lambda =\Lambda$. Relative to a $\mathbb{Z}$-basis of $\Lambda$, multiplication by $\zeta$ is represented by a unimodular matrix $A\in {\mathrm{GL}}_2(\mathbb{Z})$. Its real determinant is $|\zeta {|}^2$, while complex multiplication preserves orientation, so $\det A=1$ and $|\zeta |=1$. The Cayley-Hamilton theorem givesConsequently $2\mathrm{Re}\zeta =\mathrm{tr}(A)\in \{-2,-1,0,1,2\}$, and $\zeta$ is a root of unity of order $1$, $2$, $3$, $4$, or $6$.

A block matrix is block diagonal if all off-diagonal blocks vanish. It represents independent actions on a direct sum of vector spaces. Its determinant is the product of the diagonal-block determinants.

A Cartesian second-rank tensor has component matrix $T$ in each orthonormal basis, with transformation $T\prime =RT{R}^T$ under an orthogonal component change $R$. Both indices transform; this is why assigning arbitrary matrices independently in different bases does not define a tensor. The determinant, trace, and Frobenius inner product with another such tensor are unchanged by these basis transformations.

A closed fermion loop carries an additional minus sign from the odd Grassmann parity of the fields in the Wick theorem contraction expansion. Equivalently, integrating out Grassmann fields gives a determinant, whose logarithmic interaction expansion has the fermionic sign relative to a bosonic inverse determinant. This sign must be included in addition to propagator and vertex phases.

A real-linear map between complex vector spaces that commutes with multiplication by $i$. On $\mathbb{C}$, its real matrix is $\left(\begin{array}{cc}a& -b\\ b& a\end{array}\right)$, representing multiplication by $a+ib$. On $\mathbb{C}$, such a map with real determinant one is a rotation. In higher complex dimension, determinant one does not force the map to be a unitary operator.

Complex two-by-two matrices of determinant one form a complex three-dimensional Lie group, or a real six-dimensional Lie group. Their congruence action on Hermitian two-by-two matrices gives the Lorentz spinor double cover with kernel $\{I,-I\}$. The two Weyl spinor representations are $A$ and $(A^{†}{)}^{-1}$. Treating the group as real is essential for the second, antiholomorphic representation.

Without a symmetry excluding odd scalar powers, a real-scalar Yukawa interaction with a massive Dirac field requires a general scalar potential through degree four. For a constant background, the fermion mass becomes $M+y\varphi$, and its divergent local determinant contribution contains $(M+y\varphi {)}^4$. Linear and cubic terms are therefore generated when $M\ne 0$. A zero renormalized tadpole is a renormalization condition, not a prohibition of its counterterm. An exact discrete chiral symmetry in a massless fermion theory can instead enforce an even scalar potential.

If $f$ is a modular form on a finite-index subgroup and $\gamma$ is rational with positive determinant, then $f{|}_k\gamma$ is bounded near infinity. Choose $\sigma \in S{L}_2(\mathbb{Z})$ with $\sigma \infty =\gamma \infty$. The matrix ${\sigma }^{-1}\gamma$ is upper triangular and sends $z$ to $Az+B$ with $A>0$. Thus the existing cusp expansion of $f{|}_k\sigma$ stays bounded under this substitution. Rational conjugation of finite-index modular subgroups gives a positive period for the translate, so boundedness gives a removable singularity in its cusp parameter. Apply this to $\gamma \rho$ for every $\rho \in S{L}_2(\mathbb{Z})$ to obtain holomorphy at every cusp of the new subgroup. Vanishing at the cusps is preserved too.

A point process is determinantal with kernel $K$ relative to a specified reference measure if every correlation function of a point process is the indicated determinant. A rank-$m$ finite-rank projection kernel gives an exactly $m$-point process, whose symmetric joint density is $\det [K(x_i,x_j)]/m!$.

The top exterior power of a rank-$r$ vector bundle is its determinant line bundle. A short exact sequence of vector bundles gives a canonical isomorphism of the middle determinant with the tensor product of the other two. Locally, a subbundle frame followed by lifts of a quotient frame defines it; changing lifts only adds off-diagonal blocks. This is the determinant step in adjunction for a smooth submanifold.