How can skew-symmetric matrices be proven to be a subspace of M_{n \times n}(F)?

In summary, my friend is having trouble proving that a matrix M is skew-symmetric. I'm trying to help her, but I'm having issues with the proof myself. If any of you could offer some help, that would be much appreciated.
  • #1
RavenCpu
2
0
Hello all! I finally decided to join this forum after quite a long time of lurking haha. Anywho, to the point. A friend of mine is taking linear algebra and is having a lot of issues with the below posted problem. I'm trying to help her with it, but sadly I'm having issues with it as well haha. I would appreciate any help you all could offer me in doing this proof. I've gotten it all solved up until I have to prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]. So basically, I'm on the last step.

Prove:
A matrix M is called a skew-symmetric if [itex]M^t = -M[/itex]. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set [itex]W_1[/itex] of all skew-symmetric n x n matrices with entries from F is a Subspace of [itex]M_{n \times n}(F)[/itex]. Now assume that F is not of characteristic 2, and let [itex]W_2[/itex] be the subspace of [itex]M_{n \times n}(F)[/itex] conisting of all symmetric n x n matrices. Prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]
 
Last edited:
Physics news on Phys.org
  • #2
Rewriting it so it's legible:

Hello all! I finally decided to join this forum after quite a long time of lurking haha. Anywho, to the point. A friend of mine is taking linear algebra and is having a lot of issues with the below posted problem. I'm trying to help her with it, but sadly I'm having issues with it as well haha. I would appreciate any help you all could offer me in doing this proof. I've gotten it all solved up until I have to prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]. So basically, I'm on the last step.

Prove:
A matrix M is called a skew-symmetric if [itex]M^t = -M[/itex]. Clearly, a skew-symmetric matrix is square. Let F be a field. Prove that the set [itex]W_1[/itex] of all skew-symmetric n x n matrices with entries from F is a Subspace of [itex]M_{n \times n}(F)[/itex]. Now assume that F is not of characteristic 2, and let [itex]W_2[/itex] be the subspace of [itex]M_{n \times n}(F)[/itex] conisting of all symmetric n x n matrices. Prove that [itex]M_{n\times n}(F) = W_1 \oplus W_2[/itex]
 
  • #3
1. Prove that [itex]W_1,\, W_2[/itex] are subspaces
2. Prove [itex]W_1 \cap W_2 = \{ 0\}[/itex]
3'. Find bases [itex]\beta _1,\, \beta _2[/itex] and show that [itex]\beta _1 \cup \beta _2[/itex] is a basis for [itex]M_{n\times n}(F)[/itex]. This is easy since it's just a matter of comparing [itex]|\beta _1| + |\beta _2|[/itex] to dim(Mn x n(F)).
3''. Instead of 3', you could do the following: given any square matrix A, use your bases to compute the projection of A onto each of those subspaces, and show that the sum of the two projections is A.
3'''. You should do 3' or 3'', but another way is to prove that for any given matrix A, the equation A = (1/2)(A + At) + (1/2)(A - At) holds, and that a) A + At is symmetric, b) A - At is skew symmetric, and c) (1/2) makes sense because F is not of characteristic 2. This is the most efficient way to prove it, but it requires you to know how to decompose A in the first place. Doing the first two ways will actually teach you how, in the future, you can go about proving that some vector space is the direct sum of some subspaces. This third approach is just a handy tid-bit of knowledge, not a very instructive approach though.
 
  • #4
Thanks for rewriting it :-) I'm not too familiar with symbols and the sort in forums. It looks so clean now!

Thank you for the help! I'll give it an attempt and see what comes of it.
 

What is a skew-symmetric matrix?

A skew-symmetric matrix is a square matrix in which the elements on the main diagonal are all equal to zero, and the elements below the main diagonal are equal to the negative of the corresponding elements above the diagonal.

What is the proof for a skew-symmetric matrix?

The proof for a skew-symmetric matrix involves showing that the transpose of the matrix is equal to the negative of the original matrix. In other words, if A is a skew-symmetric matrix, then AT = -A.

Why is the main diagonal of a skew-symmetric matrix always zero?

The main diagonal of a skew-symmetric matrix is always zero because the elements on this diagonal must be equal to the negative of themselves, which is only possible if they are equal to zero.

Can a non-square matrix be skew-symmetric?

No, a non-square matrix cannot be skew-symmetric because it must have an equal number of rows and columns in order for the main diagonal to be defined and for the elements to be compared.

What are some real-world applications of skew-symmetric matrices?

Skew-symmetric matrices have many applications in physics, engineering, and computer graphics. They are used to represent quantities such as angular velocity and magnetic fields, and are also used in algorithms for image processing and computer vision.

Similar threads

  • Calculus and Beyond Homework Help
Replies
15
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
3K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
3K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
6
Views
3K
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
5K
  • Math Proof Training and Practice
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
Back
Top