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.