Skew symetric matries and basis

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In summary, the author proved that the set containing \left(\begin{array}{ccc}0 & 1 & 0 \\ -1 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right)\left(\begin{array}{ccc}0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right)and \left(\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right) is a basis for the vector space of all 3x3 skew symetric matric
  • #1
robierob12
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Lately I have been been studying basis and demension.

For a more interesting problem I wanted to see if I could find the basis of the vector space of all 3x3 skew symetric matricies.

Usually, I can find a general form for these types of problem. Such as the general form of a symetric matricie. But skew symetric matricies seem to have more than one form

[0 a b]
[-a 0 c]
[-b -c 0]

and

[0 a -b]
[-a 0 -c]
[b c 0]

I proved that this form of a skew symetric matrice is a basis

[0 a b]
[-a 0 c]
[-b -c 0]

but is it true for the vector space of all 3x3 skew symetric matricies of that form or all skew symetric matricies?
 
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  • #2
So you not see that those two forms you gave describe exactly the same set of matrices?

You proved what was a basis?
 
  • #3
I see that they are both skew symetric but becuase the general form of the two looked different they may not fit all skew semetric matricies.

Iam trying to find a basis for the vector space of all 3x3 symetric matricies.

I used this as my set

0 1 0
-1 0 0
0 0 0

0 0 1
0 0 0
-1 0 0

0 0 0
0 0 1
0 -1 0
 
  • #4
So you're trying to find a set of matrices such that every skew symmetric basis is a linear combination of them. Do you not see how to write any of the matrices in your first post in terms of those three things? Remember, you can multiply basis vectors by any scalar, including -a or -b or -c,...
 
  • #5
You seem to be concerned that while the set containing
[tex]\left(\begin{array}{ccc}0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \end{array}\right)[/tex]
and
[tex]\left(\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & -1 & 0 \end{array}\right)[/tex]

is a basis, so is the set containing
[tex]\left(\begin{array}{ccc}0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right)[/tex]
[tex]\left(\begin{array}{ccc}0 & 0 & -1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right)[/tex]
and
[tex]\left(\begin{array}{ccc}0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \end{array}\right)[/tex]

There is certainly no problem with that- any vector space has an infinite number of distinct bases!
 
  • #6
I can't believe I didnt catch that. Thanks
 

1. What is a skew symmetric matrix?

A skew symmetric matrix is a square matrix in which the transpose of the matrix is equal to its negative. In other words, for a matrix A, Aᵀ = -A. This means that all the elements on the main diagonal are equal to 0, and the elements above and below the main diagonal are the negative of each other.

2. How is a skew symmetric matrix different from a symmetric matrix?

A symmetric matrix is a square matrix in which the transpose of the matrix is equal to itself. In other words, for a matrix A, Aᵀ = A. This means that all the elements on the main diagonal are equal to each other, and the elements above and below the main diagonal mirror each other. A skew symmetric matrix, on the other hand, has elements above and below the main diagonal that are the negative of each other.

3. What is the relationship between skew symmetric matrices and cross products?

A skew symmetric matrix can be used to represent the cross product operation in three-dimensional space. The cross product of two vectors, u and v, can be written as the product of a skew symmetric matrix and the vector u: u x v = [u] x v, where [u] is a skew symmetric matrix. This relationship is useful in solving problems involving vectors and rotations in three-dimensional space.

4. How do you find the basis for a skew symmetric matrix?

The basis for a skew symmetric matrix is a set of vectors that span the space of all skew symmetric matrices. One way to find the basis is to start with a standard basis for the matrix space (i.e. the set of all n x n matrices) and then apply constraints that will result in a skew symmetric matrix. For example, setting the elements on the main diagonal to 0 and the elements above the main diagonal to negative values will result in a skew symmetric matrix.

5. What are some applications of skew symmetric matrices?

Skew symmetric matrices have applications in various fields such as physics, engineering, and computer graphics. They are commonly used in mechanics to represent moments of forces and angular velocity. In computer graphics, they can be used to represent 3D rotations and transformations. In quantum mechanics, skew symmetric matrices are used to represent spin operators. They also have applications in robotics, control theory, and image processing.

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