In linear algebra, an orthogonal transformation is a linear transformation T : V → V on a real inner product space V, that preserves the inner product...

real numbers. The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the inner product...

→ R3 is an orthogonal transformation. As a Lie group, the group of Galilean transformations has dimension 10. Two Galilean transformations G(R, v, a,...

represents a rotation (an orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant...

of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for R n {\displaystyle...

factors and orthogonal directions of scaling. It is a weighted-sum version of the dot product with positive weights—up to an orthogonal transformation. The article...

Projections (orthogonal and otherwise) play a major role in algorithms for certain linear algebra problems: QR decomposition (see Householder transformation and...

of translations, similarities, orthogonal transformations and inversions. The general form of a Möbius transformation is given by f ( z ) = a z + b c...

of an orthogonal transformation (an origin-preserving rigid transformation) with a uniform scaling (dilation). All similarity transformations (which...

Antiunitary Orthogonal transformation Time reversal Unitary group Unitary operator Unitary matrix Wigner's theorem Unitary transformations in quantum mechanics...

after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described...

to a given orthogonal transformation ( A {\displaystyle {\boldsymbol {A}}} ) if it does not change when subjected to that transformation. For invariance...

used to interpret findings of the PCA. PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a...

tridiagonal matrix is given by the continuant of its elements. An orthogonal transformation of a symmetric (or Hermitian) matrix to tridiagonal form can be...

with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation. Parallel projections...

components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional...

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension...

{\displaystyle x\mapsto A(x+b)} where A is an orthogonal matrix or the same orthogonal transformation followed by a translation: x ↦ A x + c , {\displaystyle...

coordinate transformation. The transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with...

intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal The log-polar...

) that is orthogonal to the hyperplane. The reflection of a point x {\textstyle x} about this hyperplane is the Householder transformation: x → − 2 ⟨...

to a given orthogonal transformation ( A {\displaystyle {\boldsymbol {A}}} ) if it does not change when subjected to that transformation. For invariance...

under an orthogonal transformation from O(n − 1). This shows us why we can construct Sn−1 as a homogeneous space. Oriented sphere (special orthogonal group):...

similarity transformations form the subgroup where A {\displaystyle A} is a scalar times an orthogonal matrix. For example, if the affine transformation acts...

is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of...

rotation or reflection of R k {\displaystyle \mathbb {R} ^{k}} (any orthogonal transformation, that is, any Euclidean isometry preserving 0) to the sequence...

\mathbb {R} ))} where O(n, R) is the orthogonal group. We say central fields are invariant under orthogonal transformations around 0. The point 0 is called...

on spacetime, and the group of transformations which leaves this quadratic form invariant is the indefinite orthogonal group O(3,1), a Lie group. In other...

Cartesian coordinate system. Reflection through the origin is an orthogonal transformation corresponding to scalar multiplication by − 1 {\displaystyle -1}...

every skew-symmetric matrix to a block diagonal form by a special orthogonal transformation. Specifically, every 2 n × 2 n {\displaystyle 2n\times 2n} real...