Best,j3n ny

In summary, a skew-symmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. This is different from a symmetric matrix, where the elements are equal to their corresponding elements across the main diagonal. To prove that a matrix is skew-symmetric, one can use the definition or show that it is equal to its own transpose multiplied by -1. A matrix cannot be both symmetric and skew-symmetric, and some applications of skew-symmetric matrices include physics, engineering, computer graphics, and economics.
  • #1
j3n
3
0
Hey,

I need help with thris proof.. :)

Say A is a skew-symmetric matrix. Is B=A^2 skew-symmetric, symmetric or neither. Prove your answer.



I know it's symmetric, but I'm having problems with subscriptmanship. Can anyone help me out?

Thanks,
j3n
 
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  • #2
Here's a starting point:
[tex]A=-A^\top[/tex]
 
  • #3
ny

Hi j3nny,

No problem, I can help you out with this proof. First, let's define what it means for a matrix to be skew-symmetric. A square matrix A is skew-symmetric if it satisfies the condition A^T = -A, where A^T is the transpose of A. This means that the entries on the main diagonal of A are all 0, and the entries below the main diagonal are the negatives of the corresponding entries above the main diagonal.

Now, let's consider the matrix B = A^2. Using the definition of matrix multiplication, we can write B as B = A*A. Since A is skew-symmetric, we know that A^T = -A. Therefore, we can rewrite B as B = A*(-A). Using the properties of matrix multiplication, we can also write this as B = -(A*A).

Now, since A*A is just another square matrix, we can use the definition of skew-symmetric matrices to rewrite it as A*A = -A^T. Substituting this back into our previous expression for B, we get B = -(-A^T) = A^T. This means that B is equal to its own transpose, which is the definition of a symmetric matrix.

To summarize, we have shown that if A is a skew-symmetric matrix, then B = A^2 is a symmetric matrix. Therefore, B is neither skew-symmetric nor symmetric, but rather it is a special type of matrix called a skew-symmetric matrix.

I hope this explanation helps and clarifies any confusion you had with the subscriptmanship. Let me know if you have any further questions. Good luck with your proof!


 

1. What is a skew-symmetric matrix?

A skew-symmetric matrix is a square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In other words, the matrix is equal to its own transpose multiplied by -1.

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

A symmetric matrix is a square matrix where the elements are equal to their corresponding elements across the main diagonal. In contrast, a skew-symmetric matrix has opposite signs for corresponding elements above and below the main diagonal.

3. How do you prove that a matrix is skew-symmetric?

To prove that a matrix is skew-symmetric, you can use the definition of a skew-symmetric matrix and check if all the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. Alternatively, you can also show that the matrix is equal to its own transpose multiplied by -1.

4. Can a matrix be both symmetric and skew-symmetric?

No, a matrix cannot be both symmetric and skew-symmetric. This is because a symmetric matrix has equal elements across the main diagonal, while a skew-symmetric matrix has opposite signs for corresponding elements above and below the main diagonal.

5. What are some applications of skew-symmetric matrices?

Skew-symmetric matrices are commonly used in physics and engineering, particularly in mechanics and electromagnetism. They are also used in computer graphics and computer vision algorithms for 3D transformations and image processing. Additionally, they have applications in economics, statistics, and quantum mechanics.

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