CHAPTER 04.03: UNARY MATRIX OPERATIONS: Skew symmetric matrix

So in this segment we’ll talk about a skew symmetric matrices. So in a skew symmetric matrix so you have a n by n matrix A so it is a square matrix is skew symmetric. If A I J is equal to minus A J I for all I J. So what do we mean by here it is just like a symmetric matrix but it is a skew symmetric matrix meaning that the negative element is here. So the I through J column is the negative value of the J through I column for all values of I comma J. So let’s go and look at an example and see what makes it a skew symmetric. There’s an alternate definition also. If A which is a n by n which is a square matrix is equal to minus A transpose n by n matrix. So if the transpose of the matrix is negative of the original matrix then A is skew symmetric.

So this is an alternate definition. As some people like to use this definition because it involves simply the transpose of a matrix as oppose to looking at each element which is exactly the same thing. So let’s look at an example is zero two three zero zero minus six minus two minus three plus six. This particular matrix here A which is a three by three so it’s a square matrix of the first requirement for a matrix to be skew symmetric. And then what we’re saying is that A I J is the same as minus A J I. So you’re looking at the first row second column is same as the second row first column but the only difference is negative sign. And then I have first row third column is same as the third row first column. But the only difference is the negative sign. And then I have second row third column here which is minus six is the same as the third row second column except for it is different by a negative sign.

So that makes it a skew symmetric, you got to understand that here the diagonal is important to look at. All the diagonal elements which are what you see here, zero zero and zero they have to be zero. The only way you will find out that the diagonal element, this element, is going to be negative of itself is only possible if this element this element and this element is zero. So for every skew symmetric matrix, the diagonal elements will be always zero because that’s the only way to satisfy this condition. But for any of the non-diagonal elements the only requirement is that it has the opposite sign. And that’s the end of this segment.