Skew-symmetric matrix property

In summary, the webpage discusses the relation between a DCM, angular velocity vector, and skew-symmetric matrix. Specifically, it states that the skew-symmetric matrix of the transformed angular velocity vector is equal to the transform of the skew-symmetric matrix of the original angular velocity vector. Understanding the notation and definitions is important in understanding this relation.
  • #1
ryan88
42
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This page (https://shiyuzhao.wordpress.com/2011/06/08/rotation-matrix-angle-axis-angular-velocity/), gives the following relation:

[itex]\left[R\vec{\omega}\right]_{\times}=R\left[\vec{\omega}\right]_{\times}R^{T}[/itex]

Where:

* ##R## is a DCM (Direction Cosine Matrix)
* ##\vec{v}## is the angular velocity vector
* ##[\enspace]_{\times}## represents a skew-symmetric matrix

I'm not sure where this came from. Is it some inherent property of a skew-symmetric matrix?
 
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  • #2
Looks like the transformation law for the skew symmetric matrix. To understand it, you will need to understand the notation first. It may help you to know that the statement reads: the skew symmetric matrix of the transformed angular velocity vector is equal to the transform of the skew symmetric matrix of the original angular velocity vector.

A large number of "problems" are traceable to knowing the definitions and to understanding the notation.

See if this comment helps you unravel your difficulty.
 

FAQ: Skew-symmetric matrix property

1. What is a skew-symmetric matrix?

A skew-symmetric matrix is a special type of square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the diagonal. In other words, the matrix is equal to its own negative transpose.

2. What is the defining property of a skew-symmetric matrix?

The defining property of a skew-symmetric matrix is that it is equal to the negative of its own transpose. This means that for every element aij in the matrix, there is a corresponding element aji that is equal to -aij.

3. How can I identify a skew-symmetric matrix?

A skew-symmetric matrix can be identified by checking if it is equal to the negative of its own transpose. Another way to identify it is by looking at its elements; if aij = -aji for all elements, then it is a skew-symmetric matrix.

4. What are some properties of a skew-symmetric matrix?

Some properties of a skew-symmetric matrix include: the sum of two skew-symmetric matrices is also a skew-symmetric matrix, the product of a scalar and a skew-symmetric matrix is also a skew-symmetric matrix, and the determinant of a skew-symmetric matrix is always 0 if the matrix is of odd order.

5. How are skew-symmetric matrices used in science?

Skew-symmetric matrices are commonly used in various fields of science, such as physics, engineering, and computer science. They are often used to represent physical quantities like angular velocity and magnetic fields, and are also used in algorithms for solving systems of linear equations and for data compression.

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