CHAPTER 04.02: UNARY MATRIX
OPERATIONS: Symmetric Matrix In
this segment we will talk about what a symmetric matrix is. So a square
matrix A is called symmetric if Aij is equal to Aij for all I,j.
SO the first thing which we have to understand – A is a square matrix, the
same number of rows, the same number of columns. Then the ith
row jth column has to be the same as the jth row, ith column for al the values to I,j. So that’s
when we consider the matrix to be symmetric. We can also look at it this way,
so it’s another definition. If A, if it’s a square matrix, transpose is the
same as the matrix itself. If we have a n by n
matrix, and we take the transpose of that and the same as the matric itself,
then A is symmetric. So this is an alternate definition of it, which means
the same thing as the previous definition. But some people like to look at this
definition that –hey, if I take a transpose of a square matrix and it is the
same as the original matrix, then A is considered to be symmetric. Let’s
look at an example here, let’s take a three by three matrix so it’s a square
matrix. 21, -3, -7, -3, -7, 6, 3, 2, 2, let’s suppose. SO what we are finding
out here is that we have a three by three matrix; let’s see whether it is
symmetric or not. Now, we don’t have to worry about the diagonal elements
because the row number and the column number is the same for each of these diagonal
elements, so those are going to be equal anyway. So if I switch the row
number and column number of 21, it is still the first row first column, SO
there’s no need to check that, but I need to check the elements which are not
on the diagonal. So if I look at this element right here, this is first row
second column, this is second row first column. That’s the same. This is
first row third column, this is the third row first column, they are the same. And then I have second row third column
and I have third row second column, they’re also the same. So this one is the
same as this one, this one is the same as this one, and this one is the same
as this and that’s what makes this symmetric. So
if I had to write is down, I will say that A12 which is -3 is the same as
A21. So the first row, second column is -3 is the same as the second row,
first column. A13 is -7, first row, third column is the same as A31 third
row, first column. And then I have A23 is 2, and third row, second column is
2 also, so they are equal. So that’s what makes this A matrix to be a symmetric
matrix. And that is the end of this segment. |