## Special Types of Linear Maps: Orthogonal Tensors

### Orthogonal Tensors

#### Definition

Let . is called an orthogonal tensor if .

#### Properties

Using the above definition, the following five main properties of Orthogonal Tensors can be directly deduced:

we have:

##### Property 2: Orthogonal tensors are invertible and orthogonal tensor :

The easiest way to see this is to assume that is not invertible, which implies and while . This implies that which contradicts that .

Another way to show that is invertible is to rely on the determinant function. Since , therefore, is invertible.
If then is called a proper orthogonal tensor, and If then is called an improper orthogonal tensor.

Since is invertible we can denote its inverse by which leads to . Therefore:

##### Property 3: The rows of the matrix representation of are orthonormal:

This is a direct consequence of the fact that

##### Property 4: The columns of the matrix representation of are orthonormal:

This is a direct consequence of the fact that

##### Property 5: The product of two orthogonal tensors is again orthogonal:

Indeed, let and be two orthogonal tensors, therefore:

Therefore, the product is orthogonal.

### Orthogonal Tensors in

Assume that the matrix representation of an orthogonal tensor has the following representation:

Then, using the properties above, we reach the following relations between the components:

These relationships assure the existence of an angle such that admits one of the the following two representations:

(1)

(2)

##### Proof:

Since such that: and .
Since and .

Orthogonal tensors in are either rotations or reflections. If , then is called a rotation and as shown below, represents a geometric rotation of elements of . If , then is called a reflection and as shown below, represents a geometric reflection of elements of . It is important to note that this is only true for elements of and as will be shown later, if for higher dimensions, does not necessarily represent a reflection but rather an improper rotation or “rotoinversion”.

#### Examples

The following are rotation matrices

The following are reflection matrices

#### Representation of Rotation Tensors in

Let be the angle of rotation associated with a rotation matrix . Then, given any two orthonormal vectors , admits the following representation:

(3)

##### Proof:

Since and are orthonormal and , then the following relationships hold:

Therefore, such that

Therefore, :

Therefore:

Notice that the above relationship represents a clockwise rotation of an angle . By replacing with , the counterclockwise rotation of an angle can be represented by the form:

#### Geometric Representation of Rotation Tensors in

Using the matrix representation in (1) for the following example applies a rotation of to the blue triangle to produce a red triangle. The angle is illustrated by the black arc. Notice that the matrix shown in (1) rotates the object clockwise!

### Reflection Tensors in

Reflection tensors represent the operation of reflecting elements in across a line of reflection.

#### Assertion 1:Eigenvalues of Reflection Tensors:

Reflection tensors in have the two eigenvalues 1 and -1 and the associated eigenvectors are orthogonal.

##### Proof:

It suffices to show that if is a reflection tensor in then .
Indeed:

Therefore:

Similarly,

Let be the eigenvectors associated with respectively. Therefore,

#### Assertion 2: Reflection tensors in are symmetric.

##### Proof:

This follows directly from having two orthogonal eigenvectors (See symmetric matrices).

#### Representation of Reflection Tensors in

From assertion 2 above, a reflection tensor in has two eigenvectors and associated with the eigenvalues and respectively. Therefore, and

Therefore,

(4)

#### Matrix Representations of Reflection Tensors in

In addition to the representations (2) and (4), a reflection matrix has various other representations. Let be a reflection tensor in . In a coordinate system whose basis vectors are the eigenvectors and of associated with the eigenvalues and , respectively, we denote the matrix representation of by which from (4) admits the form :

In a general coordinate system whose basis vectors are and , we first apply a coordinate transformation into the coordinate system of the eigenvectors of :

where, and are the components of the vectors and .
Notice that we can view as the perpendicular to the line of reflection since reflects . Also, and keeps its direction , so it lies on the line of reflection. The components of can be chosen such that while . In this case admits the representation:

Since and form an orthonormal basis in then admits the representation:

Where represents the geometric angle (with positive being the counter-clockwise direction) between the eigenvector and the basis vector . Therefore, admits the representation:

(5)

Comparing (5) with (2) shows that . This indicates, that the angle appearing in the reflection matrix representation (2) is equal to double the angle of inclination of the line of reflection.

In the following illustrative example the effect of varying the angle of inclination of the vector , namely on the reflection of the blue triangle is shown. The vector is illustrated by the thick black arrow, while the line of reflection is represented in green. , and are represented in black, green and red, respectively. The two equivalent matrix representations (5) and (2) are shown underneath the image.
Can you use the example below to find out the approximate inclination of the line of symmetry of the shown triangle?

### Orthogonal Tensors in

#### Assertion 1: Eigenvalues of Orthogonal Tensors in :

Proper and improper orthogonal tensors in have at least one eigenvalue that is equal to 1 or to -1 respectively.

##### Proof:

Let be a proper orthogonal tensor in , then and . Therefore:

Therefore, is an eigenvalue associated with every proper orthogonal tensor.

Similarly, if is an improper orthogonal tensor then:

Therefore, is an eigenvalue associated with every improper orthogonal tensor.

#### Representation of Orthogonal Tensors in

From the assertion above, if is an orthogonal tensor, then such that where the positive and negative signs correspond to a proper or an improper orthogonal tensor respectively.
Let form with a right hand oriented orthonormal basis set for . Then, the following relationships hold with the positive and negative sign corresponding to proper and improper orthogonal tensors respectively:

Therefore, such that

Therefore,

(6)

Where is the eigenvector associated with 1 and -1 for proper and improper orthogonal tensors respectively, and are two vectors that form with a right handed orthonormal basis set.

The following example shows a proper orthogonal (rotation) tensor in . You can vary the coordinates of the vector and the angle of rotation . The code then normalizes (shown as a blue arrow) and finds two vectors and (shown as red arrows) that are perpendicular to . Then, the proper orthogonal tensor is formed using the tensor representation in (6). The rotation is then applied to a sphere. Notice that the above form of the tensor representation rotates the sphere in a clockwise direction around .

Unlike orthogonal tensors in , an orthogonal tensor with a determinant equal to in is not necessarily associated with a reflection, but rather it represents a “rotoinversion” or an improper rotation.
The following example illustrates the action of an improper orthogonal tensor on a stack of boxes. When the angle in (6) is chosen to be zero, represents a reflection across the plane perpendicular to (The plane formed by the two red arrows). The angle represents a rotation around and thus, the action of constitutes a rotation and an inversion and hence the term “rotoinversion”. You can change the components of the vector and the angle to see the effect on the resulting transformation.

#### Matrix Representation of Orthogonal Tensors in

The tensor representation in (6) can be viewed in matrix form as follows. Given a normal vector such that , two normalized vectors and perpendicular to can be chosen. Assuming that , and form a right handed orthonormal set, then, the matrix form of a proper orthogonal tensor is given by:

(7)

The trace of a proper orthogonal matrix in is equal to .
The matrix form of an improper orthogonal tensor is given by:

(8)

The trace of an improper orthogonal matrix in is equal to .
When the angle in (8) is Degrees, the matrix represents a geometric reflection across the plane perpendicular to the vector . In this case, the matrix representation is given by:

(9)

The tensor representation (6) asserts that any rotation matrix can be viewed as a rotation around an axis . Any rotation can also be viewed using Euler’s angles as consecutive rotations around each of the basis vectors of the coordinate system. Clockwise rotations with an angles around the basis vectors , and are given by the following matrices , and , respectively:

It is important to notice that the order of rotation changes the final position of the rotated object. See the example in the rigid body rotation section.

### Problems

What values for the angle would make the matrices in (7) and (8) symmetric?.
Find the axis and angle of rotation of the rotation matrix .
Find the plane of inversion and the angle of rotation of the improper orthogonal matrices and .
Find the corresponding , and if the rotation matrix is viewed as a rotation around followed by then .
Find the corresponding , and if the rotation matrix is viewed as a rotation around followed by then .