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Optimization of a quadratic form over binary choices, here using signs $x_i\in \{-1,1\}$. Lifting $x{x}^T$ turns the objective into a matrix trace; dropping the rank-one condition gives a semidefinite relaxation of binary quadratic optimization.

For an equal-part biregular graph with part size $n$ and degree $\delta n$, let $M$ be its biadjacency matrix and $H=M-\delta J$, where $J$ is the all-ones matrix. The constant row and column sums make $H$ annihilate constants on both sides. Since $M{M}^{\mathsf{T}}={\delta }^{2}nJ+H{H}^{\mathsf{T}}$ and the two summands have zero products, their squared matrix trace splits as displayed. The bipartite four-cycle density counts the squared matrix trace of $M{M}^{\mathsf{T}}$ divided by $n^4$.

A variation of the connection one-form has $\dot{F}=D_A\dot{A}$. The Bianchi identity and graded cyclic matrix trace imply ${\partial }_t\mathrm{Tr}(F\wedge F)=2d\mathrm{Tr}(\dot{A}\wedge F)$. Thus the Chern-Weil theory four-form changes by an exact differential form, and its integral on a closed four-cycle is connection-independent. For a general unitary bundle, include the corresponding trace-product term in the Second Chern form.

Applying a scalar function separately to each matrix entry: $(f[X]{)}_{ij}=f(X_{ij})$. This differs from spectral matrix functional calculus. In particular, $\arcsin [Y]$ in Gaussian hyperplane rounding is entrywise, and its contribution to a matrix trace can be restricted to selected blocks by the support of the other matrix.

For a proper Euclidean motion of Euclidean three-space $E(x)=Ax+b$, transport the unit normal by $\tilde{N}(E(p))=AN(p)$. The corresponding shape operators are orthogonally conjugate:Their determinant and trace, hence Gaussian curvature and mean curvature, are unchanged.

For an integer $0\le k\le n$, the fantope is the convex hull of rank-$k$ orthogonal projection matrices in ${\mathbb{R}}^n$. Equivalently, it consists of real symmetric matrices with eigenvalues in $[0,1]$ and matrix trace $k$. To prove the equivalence, apply the spectral theorem for real symmetric matrices and express the eigenvalue vector as a convex combination of the zero-one extreme points of the capped simplex. The sum of the largest eigenvalues is its support function.

A gamma-matrix trace is a matrix trace of products of gamma matrices. In four spacetime dimensions, $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu })=4{\eta }^{\mu \nu }$ and $\mathrm{tr}({\gamma }^{\mu }{\gamma }^{\nu }{\gamma }^{\rho }{\gamma }^{\sigma })=4({\eta }^{\mu \nu }{\eta }^{\rho \sigma }-{\eta }^{\mu \rho }{\eta }^{\nu \sigma }+{\eta }^{\mu \sigma }{\eta }^{\nu \rho })$.

For an anti-Hermitian $SU(N)$ connection one-form, normalize the Chern-Simons three-form by $1/(8{\pi }^2)$ and use $A^g=g^{-1}Ag+g^{-1}dg$. Its change is $-d\mathrm{Tr}(dg{g}^{-1}\wedge A)/(8{\pi }^2)-\mathrm{Tr}(g^{-1}dg{)}^3/(24{\pi }^2)$. On a closed three-manifold the first term integrates to zero, while the latter is an integer with the fundamental matrix trace. Hence the integral is defined modulo integers; integer-level exponentials are invariant under large Yang-Mills gauge transformations.

The virtual similarity transformation $A^i\mapsto X{A}^i{X}^{-1}$ leaves every periodic MPS amplitude invariant by cyclicity of the matrix trace. The fundamental theorem of matrix product states says that, up to an overall phase, this is the only ambiguity for injective uniform tensors.

For a real symmetric matrix $X$ and $1\le k\le n$,Indeed $Y=Q{Q}^T$ is a rank-$k$ orthogonal projection matrix in the fantope, and projecting onto the eigenvectors of the largest eigenvalues attains the support function maximum. Equivalently, the largest matrix trace of a compression to a $k$-dimensional vector subspace is the sum of the largest eigenvalues.

For periodic $U(x,t,\lambda )$ with period $L$, solve $W_x=UW$, $W(0)=I$. The monodromy matrix is $w=W(L)$. If $U,V$ satisfy the zero-curvature condition and are periodic, then $W_t=VW-WV(0)$, hence $w_t=[V(0),w]$. The matrix trace of each power of $w$ is a first integral.