Examples of transformations

 

This implies that the eigenvalues of an isometric transformation are given by λ = exp(jφ). Then also we have ⟨ Ax , Ay ⟩ = ⟨ x m y ⟩.

When W is an invariant subspace of the isometric transformation A with dim(A) < ∞, then also W^⊥ is also invariant subspace.

 

For an orthogonal transformation O, the identity O^TO = I, so O^T = O⁻¹. If A and B are orthogonal, then AB and A⁻¹ are also orthogonal.

Let A : V → V be orthogonal with dim(V) < ∞, then A is direct orthogonal if det(A) = +1. Matrix A describes a rotation. In particular, A provides a rotation of ℝ² through angle φ, it is given by

So the rotation angle φ is determined by trace tr(A) = 2cos(φ) with 0 ≤ φ ≤ π. Let λ₁ and λ₂ be the roots of the characteristic equation. Then Re(λ₁) = Re(λ₂) = cos(φ) and λ₁ = exp(jφ) and λ₂ = exp(−jφ).

In ℝ³, λ₁ = 1, λ₂ = λ₃* = exp(jφ). A rotation over eigenspace corresponding λ₁ is given by matrix

A transformation A is called mirrored orthogonal if det(A) = −1. Vectors from E₋₁ are mirrored by A with respect to the invariant subspace E^⊥₋₁. A mirroring in ℝ² in <\( \left( \cos \left( \frac{1}{2}\,\varphi \right) , \sin \left( \frac{1}{2}\,\varphi \right) \right) \) > is given by Mirrored orthogonal transformations in ℝ³ are rotational mirroring rotations of axis < a > through angle φ and mirror plane < a >^⊥. The matrix of such transformation is given by For all orthogonal transformations in ℝ³, O(x)×O(y) = O(x×y).

ℝⁿ (n < ∞) can be decomposed in invariant subspaces with dimension 1 or 2 for each orthogonal transformation.

 

An n × n matrix U is unitary if U*U = I, the identity matrix. Its determinant is det(U) = ±1. Each isometric transformation in a finite dimensional complex vector space is unitary.

 

A square matrix A is symmetric if A^T = A. A linear transformation is symmetric if its matrix with respect to an arbitrary basis is symmetric. All eigenvalues of a symmetric transformation are real. Eigenvectors corresponding to distinct eigenvalues are orthogonal. If A is symmetric, then A^T = A = A* for any orthogonal basis. The product A^TA is symmetric if ^T is.

 

A product AB of two self-adjoint matrices A and B is self-adjoint if its commutator is zero, [A, B] = AB − BA = 0.

Eigenvalues of any self-adjoint matrix are real numbers.

 

Let the different roots of the characteristic equation of normal matrix A be β_i with multiplicities n_i. Than the dimension of each eigenspace V_i equalsn_i. These eigenspaces are mutually perpendicular and each vector x∈V can be written in exactly one way as where P_i is a projection on V_i. Subsection: Exercises

 

References