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Evaluating arbitrary monic polynomials of successive degrees $0,\dots ,m-1$ gives the same determinant as evaluating the monomials of those degrees. The basis-change matrix is triangular with diagonal entries one, so its determinant is one. This permits an orthogonal polynomial basis to replace the monomials in eigenvalue densities.

The variational characterization of trace distance turns a matrix distance into the best possible binary experiment. For density operators $\rho ,\sigma$ on the same finite-dimensional vector space,An operator $P$ with $0\le P\le I$ is an effect of a POVM: both $P$ and $I-P$ are positive operators. The Born rule assigns the first outcome probabilities $\mathrm{Tr}(P\rho )$ and $\mathrm{Tr}(P\sigma )$. Their signed gap is $\mathrm{Tr}(P\Delta )$, where $\Delta =\rho -\sigma$. Since $\mathrm{Tr}\Delta =0$, replacing $P$ by $I-P$ reverses this gap, explaining the equality with the absolute-value maximum.

One optimal $P$ is the spectral projector $P_{+}$ onto the positive eigenspaces of $\Delta$. In the spectral decomposition of $\Delta$, this choice includes all positive eigenvalues and excludes all negative eigenvalues. Their positive sum equals the trace distance by the spectral-parts formula for trace distance. Any zero eigenspace can be assigned to either outcome without changing the gap. This is the effect selected by the equal-prior Holevo–Helstrom theorem. ^[ref]

The same optimum has a classical description. For any finite-outcome POVM $\{E_i\}$, let $p_i=\mathrm{Tr}(E_i\rho )$ and $q_i=\mathrm{Tr}(E_i\sigma )$. ThenThe expression $\frac{1}{2}{\sum }_i|p_i-q_i|$ is the total variation distance of the entire outcome probability distributions; the expression for $A$ is the gap for one event. Grouping outcomes in $A$ gives the single effect $P_A={\sum }_{i\in A}{E}_i$. Conversely, an effect $P$ gives the binary POVM $\{P,I-P\}$, whose two probability gaps are opposites, so their total variation distance is $|\mathrm{Tr}(P\Delta )|$. Thus the optimum over all POVMs is already attained by one binary projective measurement. ^[ref]

For quantum state discrimination with equal prior probabilities, declare $\rho$ on outcome $P$ and $\sigma$ on outcome $I-P$. Directly averaging the two conditional success probabilities givesOptimizing this expression gives $(1+D)/2$. For the diagonal example in trace distance, $P=|0⟩⟨0|$ gives a gap $0.8-0.3=0.5$. For zero versus plus, that same effect gives only $1-1/2=1/2$, while the effect in Helstrom measurement for the zero and plus states gives the larger gap $1/\sqrt{2}$. The trace-distance-preserving binary measurement depends on the pair of preparations; a fixed measurement basis does not generally attain the trace distance for every pair.

A morphism of vector bundles over one base is a smooth map between their total spaces that commutes with projection to the base and is linear on each fiber. In local vector bundle trivializations it is multiplication by a smoothly varying matrix.

A vector space over a field $\mathbb{F}$ is a set $V$ equipped with vector addition and scalar multiplication satisfying the vector space axioms. Its vectors can represent directed displacements, lists of measurements, polynomials, or entire functions. The common structure is that adding two elements and multiplying an element by a scalar remain meaningful operations inside the same space.

For example, ${\mathbb{R}}^2$ uses coordinatewise addition and scaling. Withwe have $v=2b_1+b_2$. Moving twice along $b_1$ and once along $b_2$ reaches the same vector as five horizontal and four vertical steps. The two descriptions refer to different bases of the same space. A basis makes such descriptions unique; a spanning set makes them possible.

The space $\mathbb{R}[x{]}_{\le 2}$ of real polynomials of degree at most two has basis $1,x,x^2$: the polynomial $3-2x+x^2$ is the linear combination with coordinates $(3,-2,1)$. The zero vector here is the zero polynomial, and addition combines coefficients. A derivative is a linear map on this space; its matrix records how it acts on the three basis vectors. The same coordinate machinery therefore handles both geometric motion and operations on functions.

The choice of field matters. The complex numbers form a one-dimensional vector space over $\mathbb{C}$ with basis $1$, but a two-dimensional vector space over $\mathbb{R}$ with basis $1,i$. The question "how many independent directions?" has an answer only after the allowed scalars are specified. Likewise, polynomials of degree exactly two do not form a vector space: subtraction can remove the leading term, and the zero polynomial is absent.

The usefulness of the abstraction is that a linear map is determined by its values on a basis. Once those images are known, linear combinations give its action on every vector. The rank-nullity theorem relates the number of attainable outputs to the number of invisible input directions, while change of basis lets us choose coordinates suited to the problem. Eigenvectors, for example, choose directions on which repeated application of a linear operator becomes scalar multiplication.

A historical source is Hermann Grassmann's Die lineale Ausdehnungslehre (1844). Its full title advertises applications to other branches of mathematics, statics, mechanics, magnetism and crystallography; the original edition is digitized by ETH-Bibliothek. This gives an early setting for treating geometric quantities algebraically across different applications, rather than attributing the modern axiom list to that title alone.^[ref]

For a discrete memoryless channel with $m$ output symbols, suppose every row of the transition matrix is a permutation of every other row and all column sums agree. Such a communication channel hasThe row Shannon entropy is the same for every input, so $H(Y\mid X)$ has that fixed value for every input probability distribution. The remaining term $H(Y)$ is at most ${\log }_{2}m$ by maximum entropy on a finite alphabet.

A uniform input attains the bound. If there are $\ell$ input symbols, the total of all transition probabilities in the matrix is $\ell$; equal column sums must therefore each be $\ell /m$. Multiplication by the uniform input mass $1/\ell$ gives output mass $1/m$ in each column. Hence $H(Y)={\log }_{2}m$, which proves the displayed formula. The cyclic ternary channel capacity, binary symmetric channel capacity, and ternary symmetric channel capacity are concrete applications of this symmetry calculation.

Linearizing weakly coupled locked phases gives $0=\xi -\Omega 1-kL\theta$, where $L$ is the directed graph Laplacian. Multiplying by a normalized left null vector ${\pi }^{\mathsf{T}}$ eliminates coupling and gives the displayed weighted frequency. In a balanced network, including symmetric reciprocal coupling, $\pi =1/N$ and the frequency is the arithmetic mean. A zero diagonal in the coupling matrix alone does not imply balance. For two directed weights $p,q>0$, the weights are $(q,p)/(p+q)$.

With relaxation parameter $\omega$, weighted Jacobi updates byIts iteration matrix is $H_{\omega }=I-\omega D^{-1}A$.

Equivalently, the weighted Jacobi method takes a weighted average of the old vector and its full Jacobi method update: $x^{(k+1)}=(1-\omega )x^{(k)}+\omega x_{\mathrm{Jacobi}}^{(k+1)}$. If $Bv=\lambda v$, its iteration matrix multiplies that eigenvector by $1-\omega +\omega \lambda$. For the diagonal-4 Jacobi iteration for a three-variable tridiagonal system, the factors are $1-\omega$ and $1-\omega \pm \omega \sqrt{2}/4$; a Poisson equation smoothing parameter must not be inferred from this different matrix.

For the centered second-difference matrix with diagonal $-2$ and off-diagonals $1$, the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when $0<\omega \le 1$.

Multiplying both $A$ and $b$ by $-1$ leaves $D^{-1}A$ and the weighted Jacobi method unchanged. Thus the same formula applies to the positive second-difference matrix $T_n$ with diagonal $2$ and neighboring entries $-1$, the finite difference form of the one-dimensional Poisson equation with Dirichlet boundary conditions. Its eigenvectors have components $\sin (jk\pi /(n+1))$, and $D^{-1}{T}_n$ has eigenvalues $1-\cos (k\pi /(n+1))$. At $\omega =2/3$, upper-half modes have factors between $-1/3$ and $1/3$, while low-frequency factors remain close to one on fine grids. This explains why the weighted Jacobi method helps a multigrid method by reducing oscillatory errors, even when it is a slow solver on its own.^[ref]

Take labels $E=\{a,b,c\}$ and vectors in the real vector space ${\mathbb{R}}^2$:Here is the complete independence check for all eight subsets. The empty set is independent. Each of $\{a\},\{b\},\{c\}$ is independent because its vector is nonzero. Every pair is independent: the determinants of the two-column matrices $[v_a{v}_b]$, $[v_a{v}_c]$, and $[v_b{v}_c]$ are respectively $1$, $1$, and $-1$. The triple is dependent because $v_c-v_a-v_b=0$. ThereforeDeleting elements from any listed subset stays in this list, verifying heredity. Every exchange comparison has sizes $(0,1)$, $(0,2)$, or $(1,2)$. From the empty set, any element of the larger matroid independent set works: this covers three singleton targets and three pair targets. From any singleton to any pair, every element of the pair outside the singleton works, since every resulting pair is independent. This covers all nine singleton-pair comparisons, including disjoint ones. There are no other comparisons: all fifteen ordered size-increasing pairs satisfy exchange.

The three matroid bases are $\{a,b\}$, $\{a,c\}$, and $\{b,c\}$, all of cardinality two, with weights $13$, $14$, and $11$. The matroid greedy algorithm considers $a$, then $c$, then $b$: it admits $a$, admits $c$ because their determinant is nonzero, and rejects $b$ because the triple is dependent. The resulting matroid basis $\{a,c\}$ has weight $8+6=14$, the largest of the three totals. Every singleton can grow, so none is maximal; the three pairs are both maximal and maximum in cardinality.

The modular modular weight specifies the power $(cz+d{)}^k$ in a modular form's transformation law. Multiplication adds modular weights. At level one, the central matrix $-I$ excludes nonzero odd-weight forms; a nebentypus character at higher level changes this condition to $\psi (-1)=(-1{)}^k$.

For active rotations with $U=e^{-i\varphi J_3}{e}^{-i\theta J_2}{e}^{-i\psi J_3}$, the irreducible spin representation matrix is $D_{m{m}^{\prime }}^{(j)}=e^{-im\varphi }{d}_{m{m}^{\prime }}^{(j)}(\theta )e^{-i{m}^{\prime }\psi }$, where $d_{m{m}^{\prime }}^{(j)}=⟨j,m|e^{-i\theta J_2}|j,m^{\prime }⟩$. Its spin-one-half middle matrix is $\left(\begin{array}{cc}\cos (\theta /2)& -\sin (\theta /2)\\ \sin (\theta /2)& \cos (\theta /2)\end{array}\right)$. A $2\pi$ rotation acts by $(-1{)}^{2j}$; only integer-spin representations descend to the special orthogonal group in three dimensions.

A real Wigner matrix is a real symmetric matrix whose upper off-diagonal entries are independent centered copies of a fixed random variable, scaled to have variance of order $1/N$. Diagonal conventions may vary and should be specified separately. A common version has variance-one unscaled off-diagonal entries and either zero diagonal entries or independent diagonal entries with a bounded second moment.

The isotropic XY exchange interaction between two qubits is $E=(X\otimes X+Y\otimes Y)/2$. Its action isIndeed, $X|0⟩=|1⟩$, $X|1⟩=|0⟩$, $Y|0⟩=i|1⟩$, and $Y|1⟩=-i|0⟩$. The two terms cancel on equal bits and add on unequal bits. Thus a network Hamiltonian operator $H={\sum }_{a,b}{K}_{ab}{E}_{ab}$ preserves excitation number; in the single-excitation subspace, it is the weighted adjacency matrix, moving an excitation from $a$ to $b$ with hopping coefficient $K_{ab}$.

A matrix all of whose entries are zero. It is the additive identity among matrices of the same dimensions and represents the zero linear map in any choice of bases.