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

[Yankel]

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 !

 

[Ackbach]

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 $x=-x$?