Un-skewing a skew symmetric matrix (for want of a better phrase)

In summary, a skew symmetric matrix is a square matrix that is symmetrical about the main diagonal, but with reversed signs for elements above and below the diagonal. Un-skewing a skew symmetric matrix can be useful for certain calculations, and it can be done by replacing the negative elements below the main diagonal with their corresponding positive values. This process can also be referred to as "symmetrizing" or "antisymmetrizing" the matrix. All skew symmetric matrices can be un-skewed, but the resulting matrix may not be as relevant for certain applications.
  • #1
Trying2Learn
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TL;DR Summary
A better term for the process of creating a skew symmetric matrix
Hello

Say I have a column of components

v = (x, y, z).

I can create a skew symmetric matrix:

M = [0, -z, y; z, 0; -x; -y, x, 0]

I can also go the other way and convert the skew symmetric matrix into a column of components.

Silly question now...

I have, in the past, referred to this as "skewing a column into a skew symmetric matrix" or "unskewing the skew symmetric matrix."

Is there a better phrase to describe this? (mostly, the second one).

(And forget the algebra and the reasons... I just need the term that best describes the process.)
 
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  • #2
Representing a skew symmetric 3x3 matrix by a vector in ##\mathbb{R}^3## would be the fancy way of saying it.
 

1. What is a skew symmetric matrix?

A skew symmetric matrix is a square matrix where the elements on the main diagonal are all zero, and the elements above the main diagonal are equal in magnitude but opposite in sign to the corresponding elements below the main diagonal.

2. Why is it important to un-skew a skew symmetric matrix?

Un-skewing a skew symmetric matrix is important because it allows us to simplify calculations and perform certain operations more easily. It also helps to identify patterns and relationships within the matrix.

3. How do you un-skew a skew symmetric matrix?

To un-skew a skew symmetric matrix, we can use the transpose operation. This involves swapping the elements above and below the main diagonal, and then changing the sign of all the elements above the main diagonal. The resulting matrix will be symmetric, with all elements on the main diagonal being zero.

4. Can a skew symmetric matrix be un-skewed without using the transpose operation?

No, the transpose operation is the only way to un-skew a skew symmetric matrix. This is because the properties of a skew symmetric matrix require the elements above and below the main diagonal to be equal in magnitude but opposite in sign, and the transpose operation is the only way to achieve this.

5. What are some real-world applications of un-skewing a skew symmetric matrix?

Un-skewing a skew symmetric matrix is commonly used in fields such as physics, engineering, and computer graphics. It can be used to simplify calculations involving rotational or reflective symmetry, and to solve systems of linear equations. It is also used in image processing to perform operations such as image rotation and mirroring.

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