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Let $A$ be a real positive-definite matrix and minimize the displayed quadratic function over all of ${\mathbb{R}}^n$. Its gradient is $Ax-b$, so its only stationary point solves the linear system $A{x}_{\ast }=b$. The matrix is an invertible matrix, making this solution exist and be unique. Completing the square around that solution proves the stronger global statement:The quadratic function is strictly convex, and its level sets above the minimum are translated ellipsoids. Positivity therefore turns solving a linear system into locating the bottom of a bowl.^[ref]

For the opening matrix $A=\left(\begin{array}{cc}3& 1\\ 1& 3\end{array}\right)$, take $b=(1,1{)}^T$ and $c=0$. Then $A^{-1}=\frac{1}{8}\left(\begin{array}{cc}3& -1\\ -1& 3\end{array}\right)$, $x_{\ast }=(1/4,1/4{)}^T$, and $f(x_{\ast })=-1/4$. Every displacement $h$ raises the value by $\frac{1}{2}{h}^{T}Ah$; along the unit eigenvectors $e_{+}$ and $e_{-}$ the increases are respectively $2s^2$ and $s^2$ for displacement $s{e}_{+}$ or $s{e}_{-}$.

For a singular positive semidefinite matrix, a solution of $Ax=b$ is still a global minimizer, but adding any kernel vector gives another. If $b$ is outside the image of a linear map defined by $A$, some kernel vector $v$ has $b^{T}v\ne 0$, and $f(sv)=c-s{b}^{T}v$ is unbounded below. For an indefinite quadratic form, a negative eigenvalue gives a direction along which the quadratic term tends to $-\infty$ regardless of $b$. Thus a stationary point alone does not establish a minimum. The same energy-difference identity extends to the quadratic variational principle for a symmetric positive operator, with additional existence hypotheses in infinite dimension.

Unitary conjugation maps an operator $A$ to $UA{U}^{†}$ for a unitary operator $U$. It preserves eigenvalues, operator norms, and algebraic relations such as products and adjoints.

Two operators on Hilbert spaces are unitarily equivalent if a unitary operator maps their operator domains onto one another and intertwines their actions. Their spectra and eigenvalue multiplicities then agree. For nonnegative self-adjoint operators it suffices that the unitary maps their closed quadratic form domains onto one another and preserves the forms, since the associated operator is determined by the form.

A complex matrix $Z=-Z^T$ can be put by unitary congruence into two-by-two skew blocks and zero blocks; the magnitudes $|z_r|$ are its singular values, each repeated twice. One proof uses the antilinear map $A(v)=Z\bar{v}$. Because $Z^{†}=-\bar{Z}$, $A^2=-Z{Z}^{†}$. On a nonzero eigenspace of $Z{Z}^{†}$ with eigenvalue $s^2$, any unit vector $v$ has orthogonal partner $A(v)/s$, and $A(A(v)/s)=-sv$. These orthonormal pairs give the blocks; the orthogonal complement is invariant, so the procedure repeats. Phases of the paired vectors set the block phases.

For $n\ge 2$, the Gram matrix of $n$ unit vectors with common distinct-pair inner product $t$ is $G=(1-t)I+t11^T$. Its eigenvalues are $1+(n-1)t$ on the all-ones direction and $1-t$ on its orthogonal complement. It is a positive-definite symmetric matrix exactly when $-1/(n-1)<t<1$, permitting a linearly independent realization in ${\mathbb{R}}^n$. An explicit construction takes the columns ofThe two orthogonal projections have product zero, so $B^{T}B=G$; positive eigenvalues also make $B$ invertible. At either endpoint the corresponding eigenspace becomes a linear dependence.

At a hyperbolic equilibrium, the unstable manifold is formed by nearby trajectories approaching that equilibrium as time tends to minus infinity. It is tangent to the span of eigenvectors whose eigenvalues have positive real part. Its dimension counts departing amplitudes; translation along an autonomous trajectory can remove one parameter when comparing orbit shapes. The stable manifold instead approaches the equilibrium forward in time.

For constant-speed advection $u_t+a{u}_x=0$ with $a>0$, the explicit upwind scheme uses $U_j^{n+1}=(1-\nu )U_j^n+\nu U_{j-1}^n$, $\nu =ak/h$. For $a<0$ the neighbour must be on the other side. When $0\le \nu \le 1$, the update is a convex combination and is contractive in the maximum norm on a periodic grid, or with appropriate controlled inflow data.

On a periodic grid, von Neumann stability analysis gives $G=1-\nu +\nu e^{-i\theta }$ and $|G{|}^2=1-2\nu (1-\nu )(1-\cos \theta )$, proving the same exact contraction range in the discrete L2 norm. Boundary estimates are still required on a finite interval; stable scalar eigenvalues of a triangular boundary update alone need not give a mesh-uniform bound.

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.

In an eigenbasis of $G=X^{T}X$, ridge regression divides the corresponding component of $X^{T}Y$ by $\gamma +\lambda$. If $\gamma =0$, its eigenvector $v$ satisfies $Xv=0$, and hence $v^T{X}^{T}Y=0$ exactly. Only positive eigenvalues contribute to the limit, giving the Moore-Penrose inverse expression for the minimum-norm least-squares solution. No full-rank or sample-size assumption is needed.

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.

If $\delta D<0$, the leading interaction of two sedimenting spheres has a formal vertical equilibrium at $r_{\ast }=-3D/(2\delta )$. Choose labels with $\delta >0,D<0$ and put sphere one above sphere two. On that invariant vertical line, $\dot{r}=-\delta +\delta r_{\ast }/r$, so both sides approach $r_{\ast }$ and the radial eigenvalue is $-\delta /r_{\ast }$. A small tilt instead grows at rate $\delta /(2r_{\ast })$: the full configuration-space equilibrium is a saddle, not an attracting node. A physically valid far-field bound pair requires $r_{\ast }\gg a_1+a_2$, or $|D|\gg (a_1+a_2)|\delta |$. Reversing both driving forces reverses the vector field, interchanging radial attraction and repulsion, consistent with kinematic reversibility of Stokes flow.

At $q=a=(r-1)/(r+1)>0$, the reaction eigenvalues are $\pm i\sqrt{a}$. This linear marginal case is a weakly attracting focus, not a nonlinear center. To check it directly, put $u=(1+r)(1+x)$, $v=(1+r{)}^2(1+y)$, $h=(1+a)/2$, $\beta =(1-a)/2$, $\omega =\sqrt{a}$, and $X=x$, $Y=(hy-ax)/\omega$. The equations become $\dot{X}=-\omega Y+F_2+F_3+O(4)$, $\dot{Y}=\omega X+G_2+G_3+O(4)$, withOn the unit circle write $A=\cos \theta F_2+\sin \theta G_2$, $B=\cos \theta F_3+\sin \theta G_3$, $C=\cos \theta G_2-\sin \theta F_2$. In polar coordinates, $d\varrho /d\theta =(A/\omega ){\varrho }^2+(B/\omega -AC/{\omega }^2){\varrho }^3+O({\varrho }^4)$. The quadratic angular average vanishes. Integrating the quadratic correction over a whole cycle contributes its final squared integral, also zero, leaving the cubic average. Using the even trigonometric moments gives $⟨B⟩=a/4$ and $⟨AC⟩/\omega =(a+1)/8$. The Poincaré map is therefore $\varrho \mapsto \varrho +[2\pi /\omega ](a-1){\varrho }^3/8+O({\varrho }^4)$. Since $0<a<1$, it contracts sufficiently small positive radii and proves nonlinear asymptotic stability at the threshold.

Weak isospin labels representations of the SU(2) factor in the electroweak gauge group. A left-handed lepton doublet has $T=1/2$ with third-component eigenvalues $+1/2,-1/2$; a right-handed charged-lepton singlet has $T=0$. Along with electric charge, the third component determines the weak neutral current through $T^3-Q{\sin }^2{\theta }_W$.

A weakly nonlinear expansion writes a small disturbance as powers of an amplitude parameter while resolving its evolution on a slow time. Near a critical eigenvalue, linear detuning and nonlinear interactions enter the same perturbation order. The Fredholm solvability condition for a self-adjoint operator, or its adjoint form for a general operator, yields an amplitude equation without requiring the full higher-order correction.

For the five-point discretization of $-\Delta +c$ on the unit square, the weighted iteration eigenvalues are $1-\omega [4{\sin }^2(\pi kh/2)+4{\sin }^2(\pi lh/2)+c{h}^2]/(4+c{h}^2)$. A fixed parameter converges for every mesh when $0<\omega \le 1$, but $\omega =1$ poorly damps the highest frequencies.

For the positive scalar Laplace-Beltrami operator on a closed $d$-dimensional Riemannian manifold, the number of eigenvalues at most $\lambda$, counted with multiplicity, has the displayed leading asymptotic. The exponent determines the dimension and the coefficient determines the total volume of the Riemannian metric. There is an analogous result with fixed Dirichlet boundary conditions or Neumann boundary conditions on a compact smooth domain.

If $L$ is bounded and self-adjoint on a complex Hilbert space and $K$ is compact and self-adjoint, thenA singular Weyl sequence for $L$ stays singular for $L+K$, because compact operators send weak convergence to norm convergence and therefore $K{f}_n\to 0$. Applying the same argument with $-K$ proves the reverse inclusion. Finite-multiplicity isolated eigenvalues can move under such perturbations; the essential spectrum of a bounded self-adjoint operator is unchanged.