CHAPTER 10.10: EIGENVALUES AND EIGENVECTORS: Theorems of eigenvalues and eigenvectors Part 3 of 6     In this segment we are talking about some theorems of corresponding to eigenvalues and eigenvectors  so one of the theorems says as follows that [A] is a nxn matrix so what that means is that [A] is a square matrix then [A] transpose has same eigenvalues as [A] so what that basically means is that if you have a square matrix and you are able to find its eigenvalues then if you take its transpose then it has the same eigenvalues as the matrix so lets suppose that somebody says that this particular matrix right here [A]= 2, -3.5, 6, 3.5, 5, 2, 8, 1, 8.5 and if you take this particular matrix right here and you find its eigenvalues by following the procedure you will get 3 eigenvalues you’ll get  λ1=-1.547, λ2=12.33, λ3=4.711 so these are 3 eigenvalues which you are going to get for that particular matrix so now if somebody gave me another matrix [B] like this one so [B]= 2, 3.5, 8, -3.5, 5, 1, 6, 2, 8.5 and gave me a matrix like this one if I realize that hey this matrix [B] is simply the transpose of the [A] matrix as you can see that 2 -3.5 6 is right here 3.5 5 2 is right here 8 1 8.5 is right here so the [B] matrix here is same as the [A] transpose then the eigenvalues of [B] are the same so λ1=-1.547, so the eigenvalues of [B]  λ2=12.33, λ3=4.711and that is the end of this segment