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. |