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An automorphism of a root system $\Phi \subset E$ is an orthogonal transformation of $E$ mapping $\Phi$ onto itself. The Weyl group is a subgroup of this automorphism group; additional automorphisms can act nontrivially on a Dynkin diagram while preserving a root basis.

Every orthogonal transformation of a finite-dimensional nondegenerate quadratic space over a field of characteristic different from two is a product of at most its dimension many hyperplane reflections. In the real positive-definite case, a reflection normal to $gv-v$ sends $gv$ to $v$ whenever $gv\ne v$. Composing with that reflection fixes $v$ and leaves an orthogonal transformation of $v^{\perp }$, so induction proves the bound. Each reflection has determinant minus one, hence orientation-preserving transformations use an even number of reflections.

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.

For a real inner product space $V$, the displayed map identifies ${\Lambda }^{2}V$ with its Special orthogonal Lie algebra. It sends a wedge of distinct orthonormal basis vectors to the corresponding elementary skew-symmetric matrix, so it is an isomorphism. For an orthogonal transformation $g$, invariance of the inner product gives $\Phi (gu\wedge gv)=g\Phi (u\wedge v)g^{-1}$. Thus the exterior square of the defining representation is the Adjoint representation of the orthogonal group. Complexification identifies their weight spaces, allowing the root system to be read from the exterior-square weights and the zero-weight multiplicity.

The orthogonal transformation ${\psi }_{\pm }=({\varphi }_1\pm {\varphi }_2)/\sqrt{2}$ takes $g({\varphi }_1^4+{\varphi }_2^4+6{\varphi }_1^2{\varphi }_2^2)$ to $2g({\psi }_{+}^4+{\psi }_{-}^4)$. These are the same two decoupled Ising models expressed in rotated fields.

Every Euclidean isometry of ${\mathbb{R}}^n$ is a product of at most $n+1$ operations of reflection in a hyperplane. An isometry fixing zero is orthogonal: the polarization identity preserves inner products, and its values on an orthonormal basis determine its linear action. An orthogonal transformation is a product of at most $n$ linear reflections, by reflecting the image of the first basis vector back to that vector and inducting on its orthogonal complement. For an arbitrary isometry, first reflect its image of zero back to zero, if necessary, and apply the orthogonal result. Affine reflection hyperplanes, rather than only hyperplanes through zero, are essential here.

The normalized Hadamard transform maps a vector indexed by binary strings through entries $2^{-n/2}(-1{)}^{x\cdot y}$, where the dot product is evaluated modulo two. It is its own inverse and an orthogonal transformation, hence also a unitary operator on complex input vectors. The Hadamard gate is the $n=1$ quantum case, and the Hadamard basis is the image of the computational basis. The Walsh-Hadamard transform gives the same tensor product construction.

For two qubits, applying one Hadamard gate to each prepares the uniform quantum superpositionEach probability amplitude is $1/2$, so each computational-basis measurement outcome has probability $1/4$. Applying the Hadamard transform again reconstructs $|00⟩$: signed contributions cancel at the other three outputs. Its usefulness lies in combining simple local quantum gates into an operation on exponentially many amplitudes. Extracting useful information still requires an appropriate subsequent transformation and measurement.

Apply the preceding Householder reflection to the columns ofThe first column becomes $(-5,0{)}^T$ and the second becomes $(-11/5,2/5{)}^T$:Thus an orthogonal matrix multiplying an upper triangular matrix reconstructs $A$. The zero below the diagonal came from a geometric operation that preserves Euclidean norms and inner products.

Our positive-diagonal convention for QR decomposition is obtained by a sign adjustment. With $D=\mathrm{diag}(-1,1)$, putThen $Q^{T}Q=I$ and $\boxed{A=Q{R}_{+}}$. The intermediate sign chooses an orientation for a basis vector; the adjustment makes the diagonal positive.

For a larger matrix, the next Householder reflection acts only on the lower rows of the remaining columns, retaining the zeros already created. Store its defining vector and apply $z\mapsto z-2u(u^{T}z)$ instead of forming a dense reflection matrix. This links a simple mirror operation to efficient QR decomposition, including least-squares solutions.

A Householder transformation is a reflection that leaves every direction in a chosen hyperplane fixed and reverses its normal direction. For a real unit vector $u$, its matrix isTo understand the formula, decompose any vector into its orthogonal projection onto the normal and a perpendicular component:Then $Hx=x_{\perp }-x_{\parallel }$. Subtracting twice the normal component changes its sign while retaining everything parallel to the mirror. A nonzero normal $v$ gives the equivalent formula $H=I-2v{v}^T/(v^{T}v)$, so normalization can be deferred.

For example, take $u=(2,1{)}^T/\sqrt{5}$. Its mirror is the straight line $2x_1+x_2=0$, andThe midpoint of these two vectors is $(-1,2{)}^T$, which lies on the mirror; their connecting segment is parallel to its normal. Both vectors have Euclidean norm five.

Several useful properties follow from this decomposition. The Householder reflection is a symmetric matrix and an orthogonal matrix:Reflecting twice restores the original vector; Euclidean norms and inner products are preserved. The eigenvalue is $-1$ along $u$ and $+1$ on $u^{\perp }$, so $\det H=-1$. A reflection reverses orientation even though it preserves lengths.

The computational payoff is that the mirror can be chosen to send a whole vector onto one coordinate axis, eliminating several entries at once. Householder QR decomposition repeats this operation on the remaining parts of successive columns. For a real symmetric matrix, applying the same Householder reflection on both sides gives a matrix similarity $HAH$ that preserves symmetry and eigenvalues; this is the basis of Householder tridiagonalization.

Alston S. Householder's 1958 paper Unitary Triangularization of a Nonsymmetric Matrix develops this elimination method. Its motivation includes reducing the arithmetic needed for triangularization while retaining the conditioning advantages of orthogonal transformations. ^[ref] Over the complex numbers, $I-2u{u}^{\ast }$ for a unit $u$ is Hermitian and unitary, with the star denoting the conjugate transpose; the coordinate-axis construction below is stated for real vectors.

An orientation-reversing orthogonal transformation. In three dimensions it is a rotation about an axis followed by a reflection perpendicular to that axis. Its eigenvalues are $-1,e^{i\theta },e^{-i\theta }$, giving $\mathrm{tr}R=-1+2\cos \theta$. A plane reflection is the case $\theta =0$, while inversion through the origin is the case $\theta =\pi$.

In three-dimensional Euclidean space, a rank-three tensor invariant under every proper rotation is a scalar multiple of the Levi-Civita symbol. The distinction between proper rotations and all orthogonal transformations matters: $R_{ia}{R}_{jb}{R}_{kc}{ϵ}_{abc}=(\det R){ϵ}_{ijk}$. A polar tensor of odd rank invariant under every orthogonal transformation must vanish, since the transformation $R=-I$ changes its sign. A nonzero invariant rank-three object under reflections is instead a pseudotensor, whose transformation law includes the extra determinant factor.

For complementary vector subspaces $U,W$ of a real vector space, the rule $T(u+w)=u-w$ is a well-defined linear map satisfying $T^2=I$. It fixes $U$ pointwise and reverses $W$. In a basis formed by adjoining bases of $U$ and $W$, its matrix is $\mathrm{diag}(I,-I)$. Conversely every real linear map with $T^2=I$ has this form, with $U=\ker (T-I)$ and $W=\ker (T+I)$, as follows from $v=(v+Tv)/2+(v-Tv)/2$.

In a Euclidean space, such a linear map is an orthogonal transformation if and only if $U$ and $W$ are perpendicular. Necessity follows from $⟨u,w⟩=⟨Tu,Tw⟩=-⟨u,w⟩$; sufficiency follows by expanding the squared Euclidean norm of $u+w$. In three dimensions, a fixed plane with reversed complementary line is a possibly oblique reflection, and a fixed line with reversed complementary plane is a possibly oblique axial symmetry. Their diagonal forms alone do not imply a Euclidean reflection in a hyperplane or a Euclidean half-turn.

Let $b_1,\dots ,b_n$ be an arbitrary basis of a real inner product space, with Gram matrix $G=(⟨b_i,b_j⟩)$. If a linear map $T$ has matrix $A$ in this basis, thenConsequently $T$ is an orthogonal transformation exactly when $A^{T}GA=G$. The simpler orthogonal matrix condition $A^{T}A=I$ applies directly when the chosen basis is an orthonormal basis, so that $G=I$.

For example, take a two-dimensional vector space with weighted Euclidean inner product having $G=\mathrm{diag}(1,4)$. Thensatisfies $A^{T}GA=G$, but $A^{T}A=\mathrm{diag}(1/4,4)\ne I$. Thus it represents an orthogonal transformation even though it is not an orthogonal matrix.

The unbroken group is the stabilizer subgroup of the vacuum expectation value. For a nonzero vector vacuum expectation value, an orthogonal transformation puts $⟨\varphi ⟩=v{e}_N$. A special orthogonal transformation fixing it has the block form $\mathrm{diag}(A,1)$ with $A\in \mathrm{SO}(N-1)$. ConsequentlyThe zero vacuum expectation value leaves the entire special orthogonal group unbroken.

For a real symmetric tensor write its vacuum expectation value as a real symmetric matrix $Q$. The transformation is $Q\mapsto OQ{O}^T$. The spectral theorem for real symmetric matrices gives an orthonormal basis in whichBecause $OQ{O}^T=Q$ is equivalent to $OQ=QO$, an unbroken transformation preserves each eigenspace. The complete unbroken group is thereforeIts identity component is $H^0={\prod }_i\mathrm{SO}(m_i)$. This symmetric tensor vacuum stabilizer shows why the representation alone does not determine a unique breaking pattern:

For example a rank-one tensor $Q=v{e}_N{e}_N^T$ preserves $O(N-1)$ embedded by $A\mapsto \mathrm{diag}(A,\det A)$. Its continuous unbroken group is $\mathrm{SO}(N-1)$, but it also allows reversal of the distinguished axis. Restricting to a traceless symmetric tensor imposes ${\sum }_i{m}_i{q}_i=0$ and otherwise leaves the same classification.

For centred data, classical multidimensional scaling from Euclidean distances constructs the observation Gram matrixwhere $D_{ij}^{(2)}=\Vert x_i-x_j{\Vert }^2$. By part (i), its nonzero eigenvalues agree with those of $X^{T}X$, the matrix used for principal component analysis before dividing by a covariance normalization.

If $u_i=X{v}_i/\sqrt{{\lambda }_i}$ is the unit observation-space eigenvector, the corresponding classical multidimensional scaling coordinate column isThat is exactly the column of principal component scores. Consequently classical multidimensional scaling of Euclidean pairwise distances recovers the same reduced coordinates as principal component analysis, up to signs and arbitrary orthogonal transformations within equal-eigenvalue subspaces. Distances alone cannot fix translation or orientation. The equivalence presumes centred data and Euclidean distances; general dissimilarities need not yield a positive semidefinite Gram matrix or a corresponding data covariance matrix.

Split the mixed quadratic terms evenly across a real symmetric matrix:Its characteristic polynomial isso $c=2$. Choose the orthonormal eigenvectorsThe orthogonal transformation $\mathbf{r}=P\mathbf{R}$ givesIndeed $P^{T}P=I$ and $P^{T}AP=\mathrm{diag}(-1,-5,4)$; preserving $Q_2$ is precisely the length-preserving property of an orthogonal matrix.

The requested plane has normal $(2,1,-2)$. An orthonormal basis within it isWrite a point of that plane as $\mathbf{r}=u{e}_1+w{e}_2$. Restricting the quadratic form gives $Q_1=-u^2-(13/9)w^2$, so its section with $Q_1=-5$ isIt is an ellipse centred at the origin, with semi-axes $\sqrt{5}$ along $e_1$ and $\sqrt{45/13}$ along $e_2$. Normalizing the in-plane basis is essential: coefficients in nonorthonormal coordinates would not directly give physical lengths.

The complete surface is the one-sheet hyperboloidIt is connected, extends indefinitely in both directions along its axis $u_3$, and has an elliptical waist at $Z=0$ with semi-axes $\sqrt{5}$ and $1$. At larger $|Z|$ its transverse sections widen; its asymptotic cone is $4Z^2=X^2+5Y^2$. The specified plane is oblique to that axis, rather than the waist plane.

An orthogonal transformation to the eigenvectors preserves distance from the centre and writes the surface as $u^2+v^2+3w^2=3$. It is an ellipsoid with semi-axes $\sqrt{3},\sqrt{3},1$. ThereforeThe original $y$ axis is one of the eigen-directions with eigenvalue one, and the centre has $a_y=0$. Consequently $y^2\le 3$, attained when the other centred coordinates vanish, so $\boxed{y_{\max }=\sqrt{3}}$.

Let $T$ be the isometry. Since $T(0)=0$, it preserves norms, and the polarization identity applied to distances gives $⟨T(x),T(y)⟩=⟨x,y⟩$. For an orthonormal basis $e_1,e_2,e_3$, the vectors $T(e_i)$ are again an orthonormal basis, andExpanding in that basis proves $T(x)={\sum }_i⟨x,e_i⟩T(e_i)$. Hence $T$ is a linear map represented by an orthogonal matrix $Q$.

A reflection in a hyperplane with nonzero normal $v$ is $H_v(x)=x-2⟨x,v⟩v/⟨v,v⟩$. If $Q{e}_1\ne e_1$, choose $v=Q{e}_1-e_1$. Because $\Vert Q{e}_1\Vert =\Vert e_1\Vert$, substitution shows $H_{v}Q{e}_1=e_1$. If they agree, no reflection is needed. The resulting orthogonal transformation fixes $e_1$ and preserves its orthogonal complement. Apply the same construction there to fix $e_2$; the second normal is perpendicular to $e_1$, so the first vector remains fixed. Finally the restriction to the remaining line is either the identity or negation; in the latter case a third reflection, normal to $e_3$, fixes it. Reversing these steps writes $Q$ as a product of at most three reflections, each in a plane through the origin. This proves the three-dimensional case of the Cartan–Dieudonné theorem.

Three reflections are sometimes necessary. The map $-I$ is the product of the three coordinate-plane reflections. Its determinant is $-1$, excluding zero or two reflections. A single plane reflection fixes a two-dimensional vector subspace, whereas $-I$ fixes only zero, excluding one. Thus the bound is sharp.

By part (b), $M=S^n/\Gamma$, where the group of deck transformations acts freely by round isometries. Every round isometry is the restriction of an orthogonal transformation of ${\mathbb{R}}^{n+1}$: it preserves the inner product $⟨x,y⟩=\cos d(x,y)$, so its images of an orthonormal basis determine its linear extension.

Suppose $n$ is even. Every $A\in SO(n+1)$ has eigenvalue one. Indeed, the nonreal eigenvalues come in conjugate pairs with product one, while the remaining real eigenvalues are $\pm 1$. Their number is odd and their product is one, forcing at least one $+1$. Such a transformation fixes a unit vector. Freeness of the deck transformation action consequently implies that the determinant map $\Gamma \to \{\pm 1\}$ has trivial kernel. Thus $|\Gamma |\le 2$.

If $\Gamma$ is nontrivial with generator $A$, then $A^2=I$. Its eigenvalues are $\pm 1$, and freeness excludes the eigenvalue $+1$. Hence $A=-I$, the antipodal involution. We obtainThis proves the even-dimensional part of the classification of complete positive constant-curvature manifolds.

The conclusion fails in odd dimensions. For $n=2m-1\ge 3$, identify $S^{2m-1}$ with the unit sphere in ${\mathbb{C}}^m$. Scalar multiplication by $\zeta =e^{2\pi i/r}$, with $r\ge 3$, generates a freely acting cyclic group of isometries: ${\zeta }^{j}z=z$ for nonzero $z$ forces ${\zeta }^j=1$. The quotient is complete with constant sectional curvature one and has fundamental group $C_r$, so it is neither the simply connected sphere nor Real projective space with fundamental group $C_2$. In particular $S^3/C_3$ is a lens space counterexample.

The central problem of spectral geometry is how much of a Riemannian metric is determined by the eigenvalues, with multiplicities, of its Laplace-Beltrami operator. For a membrane with fixed boundary, the squared vibration frequencies are proportional to the Dirichlet eigenvalues. Kac's 1966 paper brought this inverse problem into focus: the full sequence of frequencies might encode a domain's shape, but recovering a few geometric quantities is much weaker than recovering an isometry class.

The first major line of development extracts geometric information from spectral sums. On a smooth closed $d$-dimensional manifold, the heat trace has the short-time expansionwhere ${\lambda }_j$ are the nonnegative eigenvalues of $P=-\Delta$ and $R$ is scalar curvature. The coefficients are heat invariants, formed by integrating local curvature expressions. They recover dimension, Riemannian volume and total scalar curvature. For smooth planar domains with the Dirichlet boundary condition,Thus area and boundary length are audible. The constant term also contains topological information through Gauss-Bonnet theorem, with corner corrections when the boundary is polygonal. The Weyl law is the corresponding leading eigenvalue-counting result. Neither the leading term nor the whole list of local heat coefficients automatically reconstructs global geometry.

In the 1970s, the wave trace introduced a complementary link with global closed geodesics. The distributionis spectral, while its singularities are governed by periodic geodesic motion. The 1975 Duistermaat-Guillemin work made this relationship precise. Its trace formula places singularities at geodesic lengths and computes their coefficients under suitable clean or nondegenerate hypotheses. When a coefficient is nonzero, the corresponding length is audible. Possible cancellation must be addressed before identifying the entire length spectrum with the singular support. This approach led to spectral rigidity questions, rather than just the recovery of integral curvature quantities.

The second main line constructs counterexamples to metric determination. A precedent was the 1964 flat-torus example: different sixteen-dimensional Euclidean lattices could give the same Laplacian eigenvalues. Subsequent lattice constructions lowered the dimension; Conway and Sloane's 1992 work exhibited four-dimensional Euclidean lattices with matching lattice theta series. For a flat torus ${\mathbb{R}}^d/\Lambda$, the spectrum is the multiset $4{\pi }^2|\xi {|}^2$ for $\xi \in {\Lambda }^{\ast }$, so equality of lattice theta series of the dual lattices gives isospectrality even when the Euclidean lattices are not related by an orthogonal transformation.

A systematic geometric construction arrived with the Sunada theorem in 1985. For a common Riemannian cover with finite isometry group $T$, almost conjugate subgroups have the same number of invariant vectors in each Laplacian eigenspace. Their freely acting quotients are therefore isospectral manifolds. This reduces an analytic equality to a finite-group representation identity. It produces many examples on surfaces and in higher dimensions, but nonconjugacy of the subgroups must be supplemented by an argument excluding extra quotient isometries. Transplantation theorem provides an explicit version: piecewise eigenfunctions are recombined by a fixed invertible linear transformation that respects the gluing and boundary conditions.

The planar problem itself was answered negatively by Gordon, Webb and Wolpert in 1992: noncongruent simply connected polygonal plane domains have equal Dirichlet spectra. This is stronger than counterexamples among abstract manifolds or flat tori, since these are genuine two-dimensional planar membranes. It does not settle every restricted version, such as asking for two domains within a specified smooth or convex class.

By the 1990s, isospectrality was also known to coexist with continuous metric variation and changes in local geometry. For example, a 1997 construction gives continuous isospectral families on products of spheres and tori whose members need not be locally isometric; in some examples the maximum scalar curvature changes. Conversely, rigidity and compactness results show that an isospectral class is not unrestricted. The 1988 surface compactness theorem makes fixed-spectrum sets of closed orientable surface metrics compact modulo diffeomorphisms. Compactness does not mean there is only one metric, or even finitely many metrics. By 2005, the subject therefore combined local heat invariants, global geodesic information, group-theoretic constructions, explicit transplantation and geometric rigidity. The spectrum determines substantial geometry, while generally failing to determine the full metric.