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An involutory matrix which is also symmetric is an orthogonal matrix, and thus represents an isometry (a linear transformation which preserves Euclidean distance). Conversely every orthogonal involutory matrix is symmetric. [3] As a special case of this, every reflection and 180° rotation matrix is involutory.
Involutory matrix: A square matrix which is its own inverse, i.e., AA = I. Signature matrices, Householder matrices (Also known as 'reflection matrices' to reflect a point about a plane or line) have this property. Isometric matrix: A matrix that preserves distances, i.e., a matrix that satisfies A * A = I where A * denotes the conjugate ...
Any involution is a bijection.. The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (x ↦ −x), reciprocation (x ↦ 1/x), and complex conjugation (z ↦ z) in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the ...
Affine involutions can be categorized by the dimension of the affine space of fixed points; this corresponds to the number of values 1 on the diagonal of the similar matrix D (see above), i.e., the dimension of the eigenspace for eigenvalue 1. The affine involutions in 3D are: the identity; the oblique reflection in respect to a plane
Any such matrix is its own inverse, hence is an involutory matrix. It is consequently a square root of the identity matrix. Note however that not all square roots of the identity are signature matrices. Noting that signature matrices are both symmetric and involutory, they are thus orthogonal.
If S is a commutative semigroup then the identity map of S is an involution.; If S is a group then the inversion map * : S → S defined by x* = x −1 is an involution. Furthermore, on an abelian group both this map and the one from the previous example are involutions satisfying the axioms of semigroup with involution.
Given a unit vector in 3 dimensions, for example (a, b, c), one takes a dot product with the Pauli spin matrices to obtain a spin matrix for spin in the direction of the unit vector. The eigenvectors of that spin matrix are the spinors for spin-1/2 oriented in the direction given by the vector. Example: u = (0.8, -0.6, 0) is a unit vector ...
The analog formula to the above generalization of Euler's formula for Pauli matrices, the group element in terms of spin matrices, is tractable, but less simple. [ 7 ] Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Pauli matrices.