CHAPTER 10.08: EIGENVALUES AND EIGENVECTORS: Theorems of eigenvalues and eigenvectors Part 1 of 6

In this segment we will talk about some of the theorems which are related to eigenvalues and eigenvectors so one of the first theorems which people talk about is if [A] is a square matrix (nxn) but it is upper triangular, lower triangular or diagonal matrix, then the eigenvalues of [A] are the diagonal entries of [A] so if you have an upper triangular matrix or lower triangular matrix or a diagonal matrix then you will find out the eigenvalues of [A] you don’t have to go through the process of finding the determinant and things like that because the eigenvalues of [A] themselves are the diagonal entries of the [A] matrix so let’s take an example if somebody gives you a lower triangular matrix like this [A]= 6, 0, 0, 3, -2, 0, 7, 6, 5 so this is a lower triangular matrix because anything above the diagonal is zero so that’s a lower triangular matrix since every element above the diagonal is zero and in this case the three eigenvalues which you will get for this particular upper triangular matrix will lower triangular matrix will be λ1= 6, λ2= -2, λ3= 5 so from this lower triangular matrix you have three eigenvalues λ1= 6, λ2= -2, λ3= 5 and that is the end of this segment n this segment we will talk about what does the word eigenvalue itself mean eigenvalue comes from the word Eigenwert it is a general word and this “Eigen” stands for characteristic and then this one the word “wert” here stands for value so that’s where the origin of the word eigenvalue comes from characteristic value it will make sense later when you when we talk about eigenvalues how we find eigenvalues what we are do is finding something called the characteristic polynomial of which we have to find the zeros of or take the take the characteristic polynomial put it equal to zero and of that equation we have to find the roots so that’s what the word eigenvalue comes from and that’s the end of this segment.