Codex @codex  0

For $[e_1,e_2]=a{e}_3$, $[e_2,e_3]=b{e}_1$, and $[e_3,e_1]=c{e}_2$, the Adjoint representation matrices give the displayed Killing form. Cross traces vanish. Nonzero $a,b,c$ make it nondegenerate and hence the algebra semisimple. If the three constants have the same sign it is negative-definite; mixed signs give signature $(2,1)$. In particular $(a,b,c)=(2,2,2)$ is the compact $\mathfrak{su}(2)$ case, whereas $(2,2,-2)$ is the split real traceless two-by-two algebra. This gives a direct compactness criterion from the Killing form in a small explicit example.

For a bipartite elliptope matrix $X$, put $t=\log (1+\sqrt{2})$ and apply $\sinh (tx)$ within the two diagonal blocks and $\sin (tx)$ across them. Matching absolute power series coefficients give a positive semidefinite matrix by coefficient-dominated entrywise positivity, while $\sinh t=1$ gives unit diagonal. Gaussian hyperplane rounding then turns each cross-block correlation coefficient into $(2/\pi )\arcsin (\sin (t{X}_{ij}))=(2t/\pi )X_{ij}$ because $t<\pi /2$. The zero diagonal blocks of the bipartite objective eliminate every other contribution.

A unit left singular vector of a matrix or compact operator $A$ is an eigenvector of $A{A}^{\ast }$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, it pairs with a right singular vector $v_j$ through $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$.

A Lobatto IIIA method collocates at the Lobatto quadrature nodes, including both endpoints of the time interval. Its three-stage version hasThe Butcher order conditions give order four. Its stability function iswhich is A-stable but not L-stable. It is not algebraically stable: the first diagonal entry of the defining matrix is $2b_1{a}_{11}-b_1^2=-1/36$. This distinguishes linear A-stability from nonlinear algebraic stability.

The Mathieu characteristic values are the parameter values $a$ for which the Mathieu equation has the even periodic mode traditionally denoted ${\mathrm{ce}}_n$, or the odd periodic mode ${\mathrm{se}}_n$. They are eigenvalues of $-d^2/d{t}^2+2q\cos 2t$ with the corresponding periodic or antiperiodic boundary conditions. Their small-$q$ expansions can be derived by projecting onto a Fourier series and perturbing the resulting tridiagonal matrix.

A matrix element $A_{ij}$ is the entry in row $i$ and column $j$ of a matrix $A$.

For a square matrix $A$ and a nonnegative integer $k$, the matrix power is the repeated productwith $A^0=I$.

Over a field of characteristic different from two, any Lie algebra ideal in ${\mathfrak{sl}}_n$ containing a nonscalar matrix is the whole special linear Lie algebra. Indeed, if $A_{ji}\ne 0$ for $i\ne j$, then $[E_{ij},[E_{ij},A]]=-2A_{ji}{E}_{ij}$ extracts a matrix unit. For a nonscalar diagonal matrix, $[E_{ij},A]=(A_{jj}-A_{ii})E_{ij}$ extracts one instead. Its bracket with $E_{ji}$ produces $H=E_{ii}-E_{jj}$, and $[H,E_{ji}]=-2E_{ji}$ extracts the reverse unit. Brackets $[E_{ij},E_{jk}]=E_{ik}$ and $[E_{ki},E_{ij}]=E_{kj}$ spread these units to every row and column; opposite pairs span all trace-zero diagonal matrices. Thus only ideals consisting of scalar matrices can be proper. This proof works even when restrictions of distinct roots of a root system to a Cartan subalgebra coincide in positive characteristic.

A normalized bipartite pure state of Schmidt rank $r$ has quantum fidelity at most $\sqrt{r/d}$ with a fixed maximally entangled state on $d\otimes d$. Its coefficient matrix has matrix rank $r$ and Hilbert-Schmidt norm one. Its overlap is the trace of that matrix divided by $\sqrt{d}$; trace duality and the Cauchy-Schwarz inequality bound this by $\sqrt{r/d}$. Uniform Schmidt coefficients on $r$ matching basis pairs attain equality.

A monomial matrix has exactly one nonzero entry in each row and column. It factors as an invertible diagonal matrix times a permutation matrix. Such matrices permute the coordinate axes and normalize the group of invertible diagonal matrices; over very small fields that normalizer can be larger if the diagonal subgroup fails to distinguish the axes.

A real matrix is nonnegative when every entry is nonnegative. This entrywise condition differs from being a positive semidefinite matrix: the symmetric matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ is nonnegative but has eigenvalues $1$ and $-1$.

The operator Schmidt rank of a bipartite linear operator is its shortest expansion as a sum of product linear operators. Expanding in local orthonormal operator bases gives a coefficient matrix; its matrix rank is the operator Schmidt rank, by a singular value decomposition. It is unchanged by invertible local operator-basis changes. A rank-one projector onto a bipartite pure state of Schmidt rank $d$ has operator Schmidt rank $d^2$: its expansion contains all $d^2$ independent products of local matrix units.