Answer: Anti-Symmetric Matrix: Necessary 0's Diagonal?

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In summary, the conversation discusses the necessity of a 0's diagonal in anti-symmetric matrices and the difference in dimensions between symmetric and anti-symmetric 2x2 matrices. The solution is that the dimension of symmetric matrices is 3 while the dimension of anti-symmetric matrices is 1, demonstrated by a matrix with a zero diagonal. The concept of anti-symmetric matrices is further explained in terms of the main diagonal and how it must satisfy $A_{ii}=-A_{ii}$. The conversation ends with the question of what numbers satisfy $x=-x$.
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Yankel
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Hello

I have a small question. Is it necessary for an anti-symmetric matrix to have a 0's diagonal ?

I have this question about the dimension of 2x2 symmetric matrices vs. dimension of anti-symmetric 2x2 matrices.

The solution is that the dim(symmetric) is 3 while dim(anti-symmetric) is 1, illustrated by a matrix with a zero diagonal.

anti-symmetric is when A=-transpose(A), will only a 0's diagonal satisfy this ?

thanks !
 
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  • #2
Think of this in terms of what must be on the main diagonal. You know that for an anti-symmetric matrix, $\mathbf{A}=-\mathbf{A}^{T}$. In an element-by-element fashion, you would write $A_{ij}=-A_{ji}$. But for elements on the main diagonal, $i=j$, and hence you'd have to have $A_{ii}=-A_{ii}$. What numbers do you know of that satisfy $x=-x$?
 

Related to Answer: Anti-Symmetric Matrix: Necessary 0's Diagonal?

1. What is an anti-symmetric matrix?

An anti-symmetric matrix is a type of square matrix where the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In other words, if aij represents an element above the main diagonal, then aji represents the corresponding element below the main diagonal and must have the opposite sign.

2. What is the significance of having 0's on the diagonal of an anti-symmetric matrix?

Having 0's on the diagonal of an anti-symmetric matrix is necessary because it ensures that the matrix will be symmetric. Since the elements below the diagonal are the negatives of the elements above, having 0's on the diagonal ensures that all elements on the diagonal will be 0, resulting in a symmetric matrix.

3. How is an anti-symmetric matrix different from a symmetric matrix?

An anti-symmetric matrix is different from a symmetric matrix in that the elements below the main diagonal are the negatives of the corresponding elements above the main diagonal. In a symmetric matrix, the elements below the main diagonal are the same as the elements above the main diagonal.

4. What are some real-world applications of anti-symmetric matrices?

Anti-symmetric matrices have various applications in physics and engineering, such as in the study of electromagnetic fields, fluid mechanics, and structural dynamics. They are also used in economics and game theory to model interactions between different agents or players.

5. How can you determine if a matrix is anti-symmetric?

A matrix is anti-symmetric if it satisfies the condition aij = -aji for all elements above the main diagonal. This means that if you switch the row and column indices of any element above the diagonal, the resulting element must be the negative of the original element. Another way to determine if a matrix is anti-symmetric is to check if its transpose is equal to the negative of the original matrix.

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