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The acoustic tensor determines the plane waves of a locally uniform incremental elastic medium. For a unit material direction $\nu$, its eigenvalues are ${\rho }_0{c}^2$ and its eigenvectors are displacement polarizations. Major symmetry of the incremental elastic moduli makes $Q$ a real symmetric matrix. Positivity on every rank-one matrix $m\otimes \nu$ makes all its wave speeds real and nonzero.

For propagation along $x$, the Alfvén right eigenvector has $\delta \rho =\delta p=\delta u_x=\delta B_x=0$, ${\mathbf{B}}_{\perp }\cdot \delta {\mathbf{u}}_{\perp }=0$ and $\delta {\mathbf{B}}_{\perp }=-s\sqrt{{\mu }_0\rho }\delta {\mathbf{u}}_{\perp }$. Its speed is $u_x+s{B}_x/\sqrt{{\mu }_0\rho }$, $s=\pm 1$. When the transverse background field vanishes, both transverse polarizations are allowed.

The unit vectors realizing an approximate eigenvalue are approximate eigenvectors: their residual tends to zero, although they need not converge to any genuine eigenvector.

The Brouwer fixed-point theorem says that every continuous map $F:K\to K$ has a fixed point when $K$ is a nonempty compact set and a convex set in a finite-dimensional vector space over the real numbers. Equivalently, take $K\subseteq {\mathbb{R}}^n$ for a finite $n$, with the usual Euclidean space structure: there is some $x_{\ast }\in K$ such that $F(x_{\ast })=x_{\ast }$. The conclusion guarantees at least one fixed point, without asserting uniqueness or telling us how quickly to find one. ^[ref]

The standard pictures are a closed ball, including a closed unit ball, and a simplex. A filled disk qualifies; its bounding circle alone does not. A probability simplex also qualifies even though its dimension is smaller than that of its ambient Euclidean space. The hypotheses have concrete meanings: being a convex set keeps every line segment between allowed points inside $K$, and compactness prevents escape or loss of a limiting point. In Euclidean space, the Heine-Borel theorem identifies compact sets with closed sets that are bounded sets. Crucially, every output must belong to the same $K$ as the input.

For a fully computable example, let $D=\{(u,v):u^2+v^2\le 1\}$ be the filled unit disk. Apply a counterclockwise quarter-turn rotation, shrink by $0.4$, and add a translation:This affine map is a continuous map. Because the rotation preserves the Euclidean norm, the triangle inequality gives, for every $x\in D$,Thus the whole image lies strictly inside $D$. More precisely, it is a disk of radius $0.4$ centred at $(0.3,0.1)$. This containment check is what makes the Brouwer fixed-point theorem applicable; a formula being a continuous map by itself would not suffice.

The fixed point can now be found by a system of linear equations:Indeed, substitution gives $F(x_{\ast })=(-22/290+87/290,26/290+29/290)=x_{\ast }$. Its Euclidean norm is $\sqrt{290}/58\approx 0.293610$, safely inside $D$. The figure shows arrows from sample inputs to their outputs; an arrow disappears at the fixed point because the displacement $F(x)-x$ vanishes there.

Repeated iterations of a map happen to succeed for this example. Starting at $x_0=(1,0)$ and setting $x_{k+1}=F(x_k)$ gives $x_1=(0.3,0.5)$, $x_2=(0.1,0.22)$, and $x_3=(0.212,0.14)$. Subtracting the fixed point equation yields $x_{k+1}-x_{\ast }=0.4R_{90}(x_k-x_{\ast })$, so the Euclidean distance to $x_{\ast }$ shrinks by exactly $0.4$ each step. This extra fact follows from the contraction mapping property and the Banach fixed-point theorem. It is additional information beyond the Brouwer fixed-point theorem.

The distinction becomes visible with two simpler continuous maps of the same disk. The pure rotation $G(u,v)=(-v,u)$ has exactly one fixed point, $(0,0)$, since $u=-v$ and $v=u$ force both to vanish. Yet iterations of a map starting at $(1,0)$ here cycle through $(0,1)$, $(-1,0)$, and $(0,-1)$ forever. Their Euclidean distance from $(0,0)$ stays one, so there is no convergence in a metric space. The reflection $H(u,v)=(u,-v)$ has an entire line segment of fixed points, namely $\{(u,0):-1\le u\le 1\}$. Starting off that line segment produces an alternating pair. Existence, uniqueness, and convergence of iterations of a map are separate questions.

Removing hypotheses changes the answer. Restrict the same quarter-turn rotation to the circle $S^1=\{(u,v):u^2+v^2=1\}$. It remains a continuous map into that compact set, but its only possible fixed point, $(0,0)$, has been removed. The circle is not a convex set. Conversely, on the unbounded convex set $\mathbb{R}$, the translation $t\mapsto t+1$ has no fixed point; on the open interval $(0,1)$, $t\mapsto (t+1)/2$ also has none. Neither domain is a compact set. The finite-dimensional closed ball statement should likewise not be extended to arbitrary infinite-dimensional vector spaces, where a closed ball need not be a compact set.

The one-dimensional version explains the connection to the intermediate value theorem. For a continuous map $f:[a,b]\to [a,b]$, put $h(t)=f(t)-t$. Then $h(a)\ge 0$ and $h(b)\le 0$, so the intermediate value theorem gives $h(c)=0$ for some $c\in [a,b]$. In higher dimensions, the Brouwer fixed-point theorem supplies a corresponding existence principle when this scalar sign argument is unavailable. Its practical value is ensuring that a proposed equation $x=F(x)$ has a solution once a suitable domain is found; it supplies no general promise of a fast algorithm.

A useful application is positive matrix eigenvector from simplex normalization: rescale the output of a strictly positive matrix to make its entries sum to one. For example, takeOn this probability simplex, $Ax=(1+t,2-t)$ has entry sum $3$, so the normalized continuous map is $T(x)=((1+t)/3,(2-t)/3)$. Both entries are positive and sum to one, keeping the image inside the probability simplex. Its fixed point equation is $t=(1+t)/3$, hence $t=1/2$. Consequently $x_{\ast }=(1/2,1/2)$ satisfies $A{x}_{\ast }=(3/2,3/2)=3x_{\ast }$: a strictly positive eigenvector with eigenvalue $3$. In larger dimensions, the Brouwer fixed-point theorem guarantees the corresponding positive eigenvector without requiring us to solve a system of linear equations in advance. The fixed point expresses preservation of relative weights, rather than preservation of their unnormalized size.

Proofs can be approached through the no-retraction theorem or Sperner's lemma. In the latter approach, increasingly fine triangulations locate approximate fixed points; compactness supplies a convergent subsequence, and continuous maps turn its limit into an exact fixed point. ^[ref]

A primary historical reference is L. E. J. Brouwer's 1911 paper Über Abbildung von Mannigfaltigkeiten, whose digitized original is hosted by Zenodo under DOI 10.1007/BF01456812. This gives a direct route to the early literature behind the Brouwer fixed-point theorem. ^[ref]

For positive-definite within-group scatter $W$ and between-group scatter $B$, directions maximizing $a^{T}Ba/(a^{T}Wa)$ solve the displayed generalized eigenvalue problem. Put $u=W^{1/2}a$; the quotient becomes the Rayleigh quotient of $W^{-1/2}B{W}^{-1/2}$. Its ordered orthonormal eigenvectors give directions orthogonal in the $W$ inner product. There are at most $g-1$ nonzero directions, where $g$ is the number of groups. These directions summarize separation for linear discriminant analysis.

An eigenvector of $A$ with eigenvalue $\lambda$ gives a homogeneous solution $t^{\lambda }\mathbf{v}$ for $t>0$. If $A$ is diagonalizable, these powers form a fundamental system. A constant vector forcing has a particular solution $t(I-A{)}^{-1}\mathbf{b}$ when $1$ is not an eigenvalue. At a resonant eigenvalue or a nontrivial Jordan block, logarithmic factors in $t$ can occur, equivalently by changing time to $\log t$ and solving a constant-coefficient linear system.

The characteristic map of a subdivision surface is the planar limit map obtained from the two independent tangent eigenvectors of its local subdivision matrix. In the usual stationary setting, a simple constant eigenvalue one, a semisimple double subdominant eigenvalue $0<\lambda <1$, strictly smaller other eigenvalues, and a regular injective characteristic map establish the standard $C^1$ tangent-plane behavior for nondegenerate data. The eigenvalue inequalities alone do not exclude a folded or singular characteristic map. For an alternating scheme these tests must use a full period of refinement.

A characteristic speed is an eigenvalue of the flux Jacobian. Along a corresponding characteristic curve, an associated wave variable obeys an ordinary differential equation obtained by projecting the governing system onto the matching left eigenvector.

For spin-projective measurements on a two-qubit state, let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of $T^{T}T$, where $T$ is its Pauli correlation tensor. Bob's sum and difference of unit axes are orthogonal and have lengths $2\cos \varphi$ and $2\sin \varphi$. Maximizing Alice's two axes aligns them with the corresponding images under $T$. Maximizing $\varphi$ then gives $2\sqrt{\Vert T\mathbf{u}{\Vert }^2+\Vert T\mathbf{v}{\Vert }^2}$ for orthonormal $\mathbf{u},\mathbf{v}$. The maximum of this quadratic-form sum is ${\lambda }_1+{\lambda }_2$, by diagonalizing $T^{T}T$ and selecting its top two eigenvectors. Thus the displayed expression is both an upper bound and attainable. The same maximum applies to the form with the sum of two absolute values, since local outcome signs can select their positive branches.

A circulant matrix applies the same rule at every position of a finite ring: moving both the input and output indices one step leaves its entries unchanged. Its first column determines the whole matrix. For a length-$N$ vector $c$, index rows and columns by $0,\dots ,N-1$ and defineThe indices use modular arithmetic. Thus each column is the previous column shifted down one place, with the last entry wrapping around to the top. Multiplication is exactly circular convolution:This makes a circulant matrix a natural model for a periodic finite difference rule or a filter acting on a periodic record. Strang's MIT lecture develops this connection between circulant matrices, circular convolution and the discrete Fourier transform.^[ref]

For example, the first column $c=(2,-1,0,-1{)}^T$ givesHere the indices wrap around modulo four. Each output compares a value with its two neighbors; this is the Graph Laplacian of a four-vertex cycle graph. A constant input gives zero, while an alternating input gives a large response. The Fourier diagonalization of a circulant matrix makes this distinction exact: its four eigenvalues are $0,2,4,2$.

The plotted discrete Fourier modes are complex vectors, shown through their real and imaginary parts. Each entry has modulus one; the Euclidean norm of each length-four vector is two. In particular, modes $k=1$ and $k=3$ have the same real part but opposite imaginary parts, so they are distinct eigenvectors with the same eigenvalue.

The useful surprise is that every length-$N$ circulant matrix shares the same basis of eigenvectors, whatever its first column. One change of basis therefore simplifies an entire family of linear maps, instead of requiring a separate eigenvalue computation for each one. Bamieh's tutorial develops the discrete Fourier transform from precisely this simultaneous diagonalization problem.^[ref]

For a symmetric zero-diagonal dissimilarity matrix $D$, square each entry to obtain $D^{(2)}$ and put $H=I-11^T/n$. Take the eigenvectors of $G=-H{D}^{(2)}H/2$ corresponding to its largest positive eigenvalues. Multiplying them by square roots of these eigenvalues gives coordinates. If $D$ contains Euclidean distances, $G$ is a centered Gram matrix, so retaining all positive eigenvalues reconstructs those distances exactly. Retaining fewer approximates the Gram matrix, not generally the sum of squared errors in raw distances. Negative eigenvalues signal failure of an exact Euclidean representation. Coordinates are unchanged in meaning by orthogonal transformations.

If $CD=DC$ and $Dx=\lambda x$, then $D(Cx)=C(Dx)=\lambda Cx$. Therefore each eigenspace of $D$ is invariant under $C$. The vector $Cx$ may be zero; invariance does not require it to be a nonzero eigenvector. This observation is the starting point for simultaneous diagonalization.

The condition $[g_Z,P_0]=0$ makes a unique global ground state an eigenvector of every filtered term, but its termwise eigenvalues can exceed the corresponding minima. For example $h_{12}=|1⟩⟨1{|}_1+|0⟩⟨0{|}_2$ and $h_{23}=|1⟩⟨1{|}_3+2|1⟩⟨1{|}_2$ commute with their sum. Their sum is uniquely minimized by $|000⟩$, yet $h_{12}$ is lower on $|010⟩$. Filtering leaves these commuting terms unchanged, so it cannot force frustration freeness.

For a two-dimensional positive-definite covariance matrix $\Sigma$, the centered covariance ellipse is the image of the unit circle under its positive definite square root of a matrix. Equivalently, it satisfies $x^T{\Sigma }^{-1}x=1$. Its semi-major and semi-minor axes have lengths given by the square roots of the larger and smaller eigenvalues, along the corresponding eigenvectors. The ellipse encodes a second-moment geometry; a probability-coverage interpretation needs additional information about the probability distribution and the chosen scale.

For a real random vector $X=(X_1,\dots ,X_p{)}^T$ with finite second moments, its covariance matrix collects all pairwise covariances:The diagonal entries are variances. The covariance matrix is a symmetric matrix and a positive semidefinite matrix, because for every real vector $a$,Thus one matrix describes the variance of every linear combination of the coordinates. It is a positive-definite matrix exactly when no nonzero linear combination is constant almost surely. A deterministic linear relation produces a zero eigenvalue.

For example,has unit coordinate variances, covariance $0.8$, and eigenvalues $1.8$ and $0.2$. Its unit eigenvectors $(1,1{)}^T/\sqrt{2}$ and $(1,-1{)}^T/\sqrt{2}$ describe directions of greatest and least projected variance. For a unit vector $a=(\cos \theta ,\sin \theta {)}^T$, direct multiplication gives $a^T\Sigma a=1+0.8\sin (2\theta )$: the variation in the sum direction is nine times that in the difference direction.

The spectral theorem supplies an orthonormal basis of these directions. This connects covariance matrices to principal component analysis: its leading eigenvector identifies the unit-direction projection with the largest variance. For a linear transformation $A$, the transformed random vector has covariance matrix $A\Sigma A^T$, so uncertainties propagate through a linear model by matrix multiplication.