A matrix is skew-symmetric if AT=−A, meaning the transpose of the matrix equals the negative of the matrix.
Let a be a diagonal element at position (i,i).
When taking the transpose of a matrix, diagonal elements remain in the same position.
For a skew-symmetric matrix, applying AT=−A:
Original diagonal element: a
After transpose: a
After applying AT=−A: −a
The element must equal its negative:
a=−a
a+a=0
2a=0
a=0
For example, consider:
A=0−2320−5−350
All diagonal elements (0,0,0) are zero.
Therefore, all diagonal elements of a skew-symmetric matrix are 0.