10. Isometries, self-adjoint mappings and unitary spaces

Part 1: problems, solutions.
Part 2: problems, solutions.

Isometries of linear euclidean spaces

A linear mapping , where , are linear euclidean spaces is an isometry if the following equivalent conditions hold

• is a linear isomorphism and preserves the inner product

• is a linear isomorphism and preserves the lengths of vectors

• takes an orthonormal basis of onto a orthonormal basis of ,
• takes any orthonormal basis of onto a orthonormal basis of ,

Orthogonal matrices

A matrix is orthogonal if .

In other words, a matrix is orthogonal if its rows (equivalently columns) form an orthonormal basis with respect to the standard inner product.

Orthogonal matrices are invertible and for such matrices.

Notice that is an isometry if and only if is orthogonal for some (equivalently, any) orthogonal bases , of respectively spaces and .

Rotations and perpendicular reflection

If is a perpendicular reflection across , and if consists of an orthonormal basis of after which we have an orthonormal basis of we get

Obviously a perpendicular reflection is an isometry.

If is a two-dimensional subspace of , then the rotation around by has the following matrix

where consists of an orthonormal basis of after which we have an orthonormal basis of . A rotation is an isometry.

It is easy to show that every isometry of a linear euclidean two-dimensional space is either a perpendicular reflection or a rotation.

Isometries of affine euclidean spaces

A mapping on affine euclidean spaces is called an isometry if one of the following equivalent conditions hold

• is an affine isomorphism and is a isometry of linear spaces,
• is an affine mapping and is orthogonal in orthonormal bases of and ,

One can prove that is an isometry iff it preserves the distance between points , i.e. for any .

Mapping preserving measure of parallelepipeds

Notice that if is a -dimensional parallelepiped and is an affine isomorphism, then is also a -dimensional parallelepiped.

Moreover, if is a -dimensional parallelepiped in a -dimensional space, then . Indeed, if , then is a basis , and is a basis . Then for an orthonormal basis , we get

Thus, an affine isomorphism on -dimensional affine euclidean spaces preserves measure of -dimensional parallelepipeds if and only if .

An endomorphism of a linear Euclidean space is self-adjoint, if for any , .

Notice that then if is an orthonormal basis of , the matrix is symmetrical! Indeed, it is so because is the -th coordinate of with respect to . Reversly, if for an orthonormal basis , matrix is symmetrical, then is self-adjoint.

It is easy to notice, that if is an eigenvector and , then . Moreover, if are eigenvectors for different eigenvalues, then . Indeed if those values are , then

so .

Also it can be proved every eigenvalue of a symmetrical matrix is real! Moreover, by an easy inductive argument using the fact of preserving perpendicularity to an eigenvector, we can prove that every symmetrical matrix (thus every self-adjoint mapping) has a basis consisting of eigenvectors. This basis is orthogonal, and can be made orthonormal. In other words, for every symmetrical matrix there exists an orthogonal matrix , such that is diagonal!

Sesquilinear and hermitian forms

The definition of the inner product which was introduced so far considers only spaces over the real numbers. Let us generalize it to the case of complex numbers. Let every space considered here be a space over complex numbers. We have to notice the following rule. The distance of a real number from zero equals . Meanwhile the distance of a complex number from zero is . This is the idea of the following definition and all subsequent definitions.

We shall say that a form is sesquilinear, if

and

for any , .

The matrix of a sesquilinear form with respect to basis of is the matrix such that

where are any vectors and and are their coordinates in basis .

A sesquilinear form is hermitian if

for any .

Hermitian inner products and unitary spaces

A hermitian form is a hermitian inner product if

for any non zero (observe that it is always a real number).

A space over along with a fixed hermitian inner product is called an unitary space.

The standard hermitian inner product in is defined by the following formula

Similarly as in the case of euclidean spaces one can define the norm, perpendicularity, orthogonal and orthonormal bases.

Hermitian matrices

A matrix is hermitian if . One can notice that a matrix of a sesquilinear form is hermitian if and only if the form is hermitian.

Mappings of unitary spaces and unitary matrices

The counterpart of an isometry is an isomorphism of unitary spaces, i.e. a linear isomorphism which preserves the hermitian product (unitary isomorphism).

It is easy to prove that a linear isomorphism of unitary spaces is unitary if and only if its matrix with respect to some orthonormal basis is such that . Such matrices are called unitary matrices.

Diagonalization of endomorphisms of an unitary space

Similarly as in the case of a space over reals, we may notice that every self-adjoint endomorphism of an unitary space has an orthonormal basis consisting of eigenvectors, and moreover the eigenvalues are real.

Another example of an mapping which is diagonalizable in an orthonormal way is an unitary automorphism. Indeed, let us prove it by induction. For one-dimensional space it is obvious. Let as assume that it is so for -dimensional spaces. Let and let be an unitary automorphism. There exists an eigenvector of , for eigenvalue (, since it is an automorphism), thus the space is -dimensional and invariant under , because if , then

We then take the basis from the induction hypothesis and add .