Special Types of Linear Maps: Skewsymmetric Tensors

Skewsymmetric Tensors Definition

Let . is called a skewymmetric tensor if .
The following properties can be naturally deduced from the definition of skewsymmetric tensors:

• In component form, the matrix representation of is such that . Therefore, the diagonal compoments are all zero.
• we have:

• is skewsymmetric. In particular, if is an orthogonal matrix associated with a coordinate transformation, then the matrix representation of stays skewsymmetric in any coordinate system.
• we have is orthogonal to . Indeed:

(1)

• Every tensor can be decomposed into two additive components, a symmetric tensor and a skewsymmetric tensor

The following is an example of the matrix representation of a skew symmetric tensor :

Skewsymmetric Tensors in

Properties

Skewsymmetric tensors in represent the instantaneous rotation of objects around a certain axis. In fact, for every skewsymmetric tensor , there exists a vector , such that . In other words, the action of on any vector can be represented as the cross product between a fixed vector and . We will show this by first looking at one of the eigenvalues of a skewsymmetric tensor:

Assertion 1:

is an eigenvalue for any skewsymmetric tensor

Proof:

Note that this result applies to any vector space with dimensions when is odd.
Since the characteristic function of , namely produces a polynomial of a third degree, it has at least one real eigenvalue. Therefore, there exists a corresponding eigenvector, say . However, from (1), is orthogonal to which means that is orthogonal to , but cannot be the zero vector, so, has to be the zero vector, therefore, . These statements can also be written as follows:

Notice that this implies that is not invertible!

Assertion 2:

The action of a skewsymmetric tensor is equivalent to the cross product operation in the following manner: If is a skewsymmetric tensor and is the normalized eigenvector associated with the eigenvalue . If form a right handed orthonormal basis set in , then :

where

Proof:

First we show that and . To show this, we will use the fact that and that .
Indeed, since form an orthonormal basis set, then, such that , the components and can be found by taking the dot product between and the vectors and .

Therefore:

Similarly,

Finally we show that . Indeed, since form a right handed orthonormal basis set, then, such that .
Therefore:

Also:

Therefore:

(2)

The vector is called the axial vector of .

The matrix representation of a skewsymmetric tensor in

In an arbitrary coordinate system defined by the orthnormal basis set , the matrix representation of a skewsymmetric tensor has the following form:

The axial vector of adopts the following form (Why?):

The relationship between the skewsymmetric tensors and rotations in

Skewsymmetric matrices with real number entries are the slopes of real orthogonal matrices around the identity matrix, i.e., skewsymmetric matrices can be considered as infinitesimal rotations.
For example, consider the following rotation matrix:

Where, is time. The matrix is a function of time and describes the counterclockwise rotation of objects in around the vector with an angular velocity .
The time derivative of , namely has the form:

When , is the identity matrix and is then a skewsymmetric matrix:

describes the velocity of counterclockwise rotation around the axial vector with an angular velocity .

We can now generalize this for every rotation matrix.

Assertion:

For very small rotations, the rate of change of a rotation tensor is represented by a skewsymmetric tensor

Proof:

Let be a rotation tensor that varies as a function of time. Assume also, that at , the rotation angle , i.e., .
Then:

Where is the time derivative of . For small rotations, or in other words when and :

Therefore, is a skewsymmetric tensor.
Notice that the same proof applies if instead we take the derivative of with respect to , i.e., is a skewsymmetric tensor at .

The above asserts that the time derviative of a rotation tensor at small rotations is a skewsymmetric tensor.
We will now look at the skewsymmetric tensors themselves to show that:

Assertion:

Every skewsymmetric tensor represents the speed of rotation (the rate of change of a rotation matrix).

Proof:

The relationship (2) asserts that the action of a skewsymmetric tensor on a vector corresponds to the operation .
Recall that if an object is rotating counterclockwise with an angular velocity around a unit vector , then the velocity vector of each point (represented by a vector ) on the object will be equal to .
i.e., a skewsymmetric tensor describes the angular velocity around its real eigenvector . Recall that this eigenvector corresponds to the eigenvalue .

Consider, the counterclockwise infinitesimal rotation around a normalized vector with angular velocity . The infinitesimal angle of rotation is equal to where is an infinitesimal time duration.
The infinitesimal rotation is then described by the skewsymmetric tensor . The new position of every vector after rotating would be equal to its original position plus a small increment corresponding to its infintesimal angular rotation. Thus, the vector is transformed into the vector . Setting

(3)

then, is indeed a rotation tensor since

The relationship (3) asserts that every skewsymmetric tensor represents the rate of change of a rotation matrix .