Isometries on Real Inner Product Spaces Every Isometry Is Normal

Isometries on Real Inner Product Spaces Every isometry is normal

Suppose S ∈ L(V). Then the following are equivalent: S is an isometry. S∗S = I. SS∗ = I.

Review of Isometries

Deﬁnition: isometry

An operator S ∈ L(V) is called an isometry if kSvk = kvk for all v ∈ V. In other words, an operator is an isometry if it preserves norms. Review of Isometries

Deﬁnition: isometry Every isometry is normal

An operator S ∈ L(V) is called an Suppose S ∈ L(V). Then the following isometry if are equivalent: kSvk = kvk S is an isometry. ∗ for all v ∈ V. S S = I. ∗ In other words, an operator is an SS = I. isometry if it preserves norms. The matrix of this operator with respect to the standard basis is cos θ − sin θ . sin θ cos θ

If θ is not an integer multiple of π, then no nonzero vector of R2 gets mapped to a scalar multiple of itself, and hence the operator has no eigenvalues.

Examples of Isometries

Example: Let θ ∈ R. Then the operator on R2 of counterclockwise rotation by an angle of θ is an isometry, as is geometrically obvious. If θ is not an integer multiple of π, then no nonzero vector of R2 gets mapped to a scalar multiple of itself, and hence the operator has no eigenvalues.

Examples of Isometries

Example: Let θ ∈ R. Then the operator on R2 of counterclockwise rotation by an angle of θ is an isometry, as is geometrically obvious. The matrix of this operator with respect to the standard basis is cos θ − sin θ . sin θ cos θ Examples of Isometries

If θ is not an integer multiple of π, then no nonzero vector of R2 gets mapped to a scalar multiple of itself, and hence the operator has no eigenvalues. Proof Suppose (a) holds, so S is an isometry. There is an orthonormal basis of V with respect to which S has a block diagonal matrix such that each block is a 1-by-1 matrix or a 2-by-2 matrix of the form a −b , b a with b > 0. If λ is an entry in a 1-by-1 matrix along the diagonal of the matrix of S, then λ = 1 or λ = −1. Because a2 + b2 = 1 and b > 0, we can choose θ ∈ (0, π) such that a = cos θ and b = sin θ. Thus the matrix has the required form, proving that (a) implies (b).

Description of Isometries on Real Inner Product Spaces

Description of isometries when F = R

Suppose V is a real inner product space and S ∈ L(V). Then the following are equivalent: (a) S is an isometry. (b) There is an orthonormal basis of V with respect to which S has a block diagonal matrix such that each block on the diagonal is a 1-by-1 matrix containing 1 or −1 or is a 2-by-2 matrix of the form cos θ − sin θ , sin θ cos θ with θ ∈ (0, π). There is an orthonormal basis of V with respect to which S has a block diagonal matrix such that each block is a 1-by-1 matrix or a 2-by-2 matrix of the form a −b , b a with b > 0. If λ is an entry in a 1-by-1 matrix along the diagonal of the matrix of S, then λ = 1 or λ = −1. Because a2 + b2 = 1 and b > 0, we can choose θ ∈ (0, π) such that a = cos θ and b = sin θ. Thus the matrix has the required form, proving that (a) implies (b).

Description of Isometries on Real Inner Product Spaces Proof Suppose (a) holds, so S is an Description of isometries when F = R isometry.

Suppose V is a real inner product space and S ∈ L(V). Then the following are equivalent: (a) S is an isometry. (b) There is an orthonormal basis of V with respect to which S has a block diagonal matrix such that each block on the diagonal is a 1-by-1 matrix containing 1 or −1 or is a 2-by-2 matrix of the form cos θ − sin θ , sin θ cos θ with θ ∈ (0, π). If λ is an entry in a 1-by-1 matrix along the diagonal of the matrix of S, then λ = 1 or λ = −1. Because a2 + b2 = 1 and b > 0, we can choose θ ∈ (0, π) such that a = cos θ and b = sin θ. Thus the matrix has the required form, proving that (a) implies (b).

Description of Isometries on Real Inner Product Spaces Proof Suppose (a) holds, so S is an Description of isometries when F = R isometry. There is an orthonormal basis of V with respect to which S has Suppose V is a real inner product space and a block diagonal matrix such that each S ∈ L(V). Then the following are equivalent: block is a 1-by-1 matrix or a 2-by-2 matrix of the form (a) S is an isometry. a −b (b) There is an orthonormal basis of V with , b a respect to which S has a block diagonal matrix such that each block on the diago- with b > 0. nal is a 1-by-1 matrix containing 1 or −1 If λ is an entry in a 1-by-1 matrix along or is a 2-by-2 matrix of the form the diagonal of the matrix of S, then λ = 1 or λ = −1. Because a2 + b2 = 1 cos θ − sin θ , and b > 0, we can choose θ ∈ (0, π) sin θ cos θ such that a = cos θ and b = sin θ. with θ ∈ (0, π). Description of Isometries on Real Inner Product Spaces Proof Suppose (a) holds, so S is an Description of isometries when F = R isometry. There is an orthonormal basis of V with respect to which S has Suppose V is a real inner product space and a block diagonal matrix such that each S ∈ L(V). Then the following are equivalent: block is a 1-by-1 matrix or a 2-by-2 matrix of the form (a) S is an isometry. a −b (b) There is an orthonormal basis of V with , b a respect to which S has a block diagonal matrix such that each block on the diago- with b > 0. nal is a 1-by-1 matrix containing 1 or −1 If λ is an entry in a 1-by-1 matrix along or is a 2-by-2 matrix of the form the diagonal of the matrix of S, then λ = 1 or λ = −1. Because a2 + b2 = 1 cos θ − sin θ , and b > 0, we can choose θ ∈ (0, π) sin θ cos θ such that a = cos θ and b = sin θ. with θ ∈ (0, π). Thus the matrix has the required form, proving that (a) implies (b). Euclid Explaining Geometry Linear Algebra Done Right, by Sheldon Axler