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For a homogeneous Keller--Segel model with nonnegative chemical production, linearized cell diffusion $D_2>0$, attraction coefficient $D_1>0$, and chemical diffusion $D_{\rho }>0$, put $c=\kappa -a_0{f}_0^{\prime }$ and $\kappa =(\rho k(\rho ){)}^{\prime }{|}_{{\rho }_0}$. A Fourier mode with squared wavenumber $z$ has determinant $z[D_2(c+D_{\rho }z)-D_1{f}_0]$. If $c\ge 0$, a growing mode on the infinite plane exists exactly when $D_1{f}_0>D_{2}c$, and $0<z<(D_1{f}_0-D_{2}c)/(D_2{D}_{\rho })$ is the unstable band. If $c<0$, the homogeneous chemical mode is already unstable. Dividing the threshold by $D_2\kappa$ to obtain a sum of two ratios is valid only when $\kappa >0$. Positivity of $k$ does not guarantee positivity of $(\rho k{)}^{\prime }$.

For paraxial rays, the Lagrange optical invariant is $H=n(y_1{u}_2-y_2{u}_1)$. Lossless paraxial ray transfer preserves this quantity: in coordinates $(y,nu)$, free propagation and a thin lens act by matrices with determinant one, which preserve the oriented area of two ray vectors. Consequently a full angular optical slit width $\theta$ at an optical pupil diameter $D$ satisfies $D\theta =A_{\mathrm{cam}}p/f_{\mathrm{cam}}$ for its image width $p$ and optical camera optical pupil width $A_{\mathrm{cam}}$ in the same meridional plane.

For distinct ${\ell }_i$ and $c_j$, expand the exponential entries in powers of $t$. Nonzero determinant terms first occur at the distinct powers $0,1,\dots ,m-1$. The coefficient is the product of their two Vandermonde determinants divided by the factorial product. Ratios of such alternants yield the Weyl dimension formula by taking a torus character to the identity.

If $\dot{\Phi }=A(t)\Phi$ and $\Phi (0)=I$, differentiation of the determinant gives $\det \Phi (t)=\exp ({\int }_0^t\mathrm{tr}A(s)ds)$. For a planar periodic orbit this gives the nontrivial Floquet multiplier, since the tangent multiplier is one.

A lower triangular matrix has zero entries above its main diagonal. Its determinant is the product of its diagonal entries, and an invertible lower triangular linear system is solved by forward substitution.

The determinant is a smooth Lie group homomorphism from the real general linear group to ${\mathbb{R}}^{\times }$. Its derivative at the identity is the trace, and the flow-defined exponential on ${\mathbb{R}}^{\times }$ is the ordinary scalar exponential. Naturality of the Lie group exponential therefore proves the identity without diagonalizability assumptions. The series ${\sum }_{k\ge 0}{t}^k{A}^k/k!$ is the flow-defined matrix exponential because it solves $E^{\prime }=EA$, $E(0)=I$, and has inverse $E(-t)$.

A monomial alternant is the determinant formed from powers with an integer exponent tuple $\ell$. It is a polynomial when the exponents are nonnegative, and otherwise a Laurent polynomial. Equal exponents give equal columns and zero determinant; interchanging exponents changes its sign. Ordered strictly decreasing nonnegative exponents give a basis of alternating polynomials, by grouping monomials into permutation orbits.

A multilinear map $F:V_1\times \cdots \times V_r\to W$ is a map between vector spaces that is linear in each argument when all other arguments are fixed. For example the evaluation $(\omega ,v)\mapsto \omega (v)$ is bilinear, and the determinant is multilinear in its columns. The universal property of a tensor product replaces this separate linearity by a single linear map from $V_1\otimes \cdots \otimes V_r$.

In odd dimension, the map $A\mapsto (\det A)A$ is a surjective group homomorphism from the orthogonal group to the special orthogonal group, with kernel of a group homomorphism $\{I,-I\}$. Scalar matrices commute, and $\det [(\det A)A]=(\det A{)}^{2m+2}=1$. Together with the determinant sign, it identifies $O(2m+1)$ with the direct product of groups $SO(2m+1)\times C_2$. In even dimension, scalar multiplication by $\det A$ does not repair the determinant, so this construction does not give the same splitting.

For real monic orthogonal polynomials $p_j$ with squared weighted norms $h_j$ under a positive weight $w$, the functions $p_j\sqrt{w/h_j}$ form an orthonormal set for the unweighted reference measure. Their finite-rank projection kernel turns the squared weighted Vandermonde determinant into a kernel determinant. Normalization on the full labelled configuration space is $1/m!$.