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For a convex cone $D$ in an inner product space, use the nonnegative-pairing conventionHere the pairing is the Frobenius inner product on real symmetric matrices. Let $C_0$ be the conic hull of the nonnegative rank-one matrices $x{x}^T$, and let $C=\overline{C_0}$. For $A\in K$, $⟨A,x{x}^T{⟩}_F=x^{T}Ax\ge 0$. This extends to conic combinations, and by continuity to their limits. Hence $C\subseteq K^{\ast }$.

For the converse, if $B\notin C$, separation from a closed convex cone supplies a symmetric $H$ withIn particular $x^{T}Hx\ge 0$ for every $x\ge 0$, so $H\in K$. The negative pairing then excludes $B$ from $K^{\ast }$. ThereforeThe same generator test gives $C^{\ast }=K$. This is the duality of copositive and completely positive cones; the next argument removes the closure.

The Krivine rounding constant has $t=c_K\pi /2=\log (1+\sqrt{2})<\pi /2$. Since $|X_{ij}|\le 1$, the principal real inverse sine satisfieson every cross-block entry. The matrix $A$ has zero diagonal blocks, so those cross blocks are the only contributors to its Frobenius inner product with $\arcsin [Y]$. Explicitly,No analogous identity is needed for the diagonal blocks involving $\sinh$.

Combining with the preceding expectation formula and optimality of $X$ gives the bipartite sign rounding boundAt least one feasible rounded outcome attains at least this expectation. This is an expectation-based approximation guarantee for the specific bipartite sign problem; no claim is made that $c_K$ is the largest possible constant. The inequalities remain valid when the optimal objective is zero.

By the definitions of matrix multiplication, transpose and trace,Relabelling the summed indices givesthe Frobenius inner product of these real matrices.

The tensor contraction of a symmetric tensor with an antisymmetric one vanishes. Rename the two summed indices and use their opposite exchange symmetries:Hence $R_{ij}{A}_{ij}=0$. Using the decomposition from part (iii) givesThis is orthogonality of the real symmetric and antisymmetric matrix subspaces under the Frobenius inner product. It is a summed scalar identity, not an assertion about the unsummed products of individual entries.

Relabel the two dummy indices and use the respective symmetries:Thus the symmetric tensor and antisymmetric tensor subspaces are orthogonal in the Frobenius inner product.

The orthogonal projection $P=Q{Q}^T$ givesThe first summand has every column in the column space of $Q$, and the second has every column in its orthogonal complement. They therefore have zero Frobenius inner product. The Pythagorean identity, together with $Q^{T}Q=I_r$, gives the stronger exact formulaConsequentlywith equality exactly when $M=Q^{T}A$. Thus solving the linear least-squares problem for each column chooses the best coefficients in the prescribed column space.

Take $N=4p-1$, $|A|=|B|=2p-1$ and sign vectors $u_A=21_A-1$. The Frobenius inner product of two outer products is $(u_A\cdot u_B{)}^2$, proving the formula. The squared Euclidean distance is largest precisely when $|A\cap B|=p-1$, where the inner product is minus one. The common diagonal can be deleted, leaving an isometric representation in dimension $\left(\genfrac{}{}{0ex}{}{N}{2}\right)$ if off-diagonal coordinates are multiplied by $\sqrt{2}$. The Frankl-Wilson theorem then bounds each smaller-diameter class.

If an array specified in each orthonormal basis has invariant Frobenius inner product with every Cartesian second-rank tensor, it transforms as such a tensor. For a component change $R$, invariance says $(R^{T}A\prime R-A):B=0$ for every matrix $B$. Nondegeneracy of the Frobenius inner product forces the difference to vanish. Testing every tensor, rather than a single selected tensor, is essential.

For a proper cone $K$, suppose a point satisfies $Ax-b\in \mathrm{int}K$ and $p_{\ast }$ is finite. This generalized Slater condition gives strong duality and attainment of the conic dual problem's optimum. It does not by itself assert primal attainment.^[ref]

The two worked examples have strict primal points: $(u,v)=(2,2)$ has slack $(2,2,1)$ in the interior of the nonnegative orthant, and $(t,z)=(2,0)$ has slack $(2,1,0)$ in the interior of the second-order cone. Both optima were already certified by explicit feasible pairs. Slater's condition explains why a matching optimal dual variable can be expected under these hypotheses; checking a supplied zero-gap pair needs only weak duality.

Changing cone geometry preserves the bound mechanism. For a positive semidefinite cone, the dual cone pairing is the Frobenius inner product of two matrices. This connects the same certificate idea to semidefinite programming, while the central path uses strictly interior primal and dual points to approach vanishing gap.

In 1995, Lieven Vandenberghe and Stephen Boyd presented a primal-dual interior-point method for convex optimization with matrix inequalities. Their algorithmic study included exploiting problem structure and allowing incomplete Conjugate gradient method solves within an iteration. This illustrates how the primal-dual viewpoint guides computation as well as checking a final answer.^[ref]

Let $S$ be a real symmetric matrix and $A$ a real antisymmetric matrix of the same size. Invariance of the trace under transpose and the cyclic trace identity giveHence $\mathrm{tr}(SA)=0$. Equivalently these two matrices are orthogonal for the Frobenius inner product, since $⟨S,A{⟩}_F=\mathrm{tr}(S^{T}A)=\mathrm{tr}(SA)$. Together with the unique symmetric and antisymmetric parts of a matrix, this is an orthogonal decomposition of the real square matrices.