Help Diagonal matrix similar to upper triangular matrix?

[thebuttonfreak]

I have been thinking about this for a week and I simply can't get anywhere.

consider a 4x4 diagonal matrix such that if a11 > a22 > a33 > a44, (0's everywhere else)

then it is similar to another matrix with the same diagonal but 0's in upper triangular part an any numbers in the lower triangular part.

These seem to be similar regardless of whether a11 > a22 > a33 > a44


help!

 

[Defennder]

Have you learned the concept of matrix diagonalization yet? It's a lot easier to understand why the 2 matrices are similar if you already have. BTW, I think you meant lower triangular matrix in the thread title, but it is true that under your specified conditions, the triangular matrix, be it upper or lower, is indeed similar to that of the diagonal one.

Assuming that you know what is matrix diagonalization, note that the eigenvalues of a triangular matrix (denoted A) are the values of the diagonal entries. With that in mind, and if the diagonal entries are distinct (a sufficient but not necessary condition), then the matrix is diagonalizable. And if the matrix is diagonalizable, note that the diagonal matrix so formed (denoted D) would have the eigenvalues of the triangular matrix as its diagonal entries. And therefore, since (P^-1)AP=D , by diagonalizability, the two matrices D and A are similar.

I'm still thinking of how to explain it without the concept of diagonalizability.

 

[tacman]

Hi there,

I understand that you have been struggling with understanding the concept of diagonal matrices and their similarity to upper triangular matrices. Let me try to break it down for you.

Firstly, a diagonal matrix is a square matrix where all the elements outside the main diagonal (the diagonal from top left to bottom right) are zero. In other words, it is a matrix with non-zero elements only on the main diagonal.

On the other hand, an upper triangular matrix is a square matrix where all the elements below the main diagonal are zero. This means that the non-zero elements can be on the main diagonal and above it.

Now, when we talk about similarity between matrices, we mean that they can be transformed into each other through a series of elementary row operations (such as swapping rows or multiplying a row by a constant).

In the example you provided, both the diagonal matrix and the upper triangular matrix have the same non-zero elements on the main diagonal. This means that they can be transformed into each other through elementary row operations. Therefore, they are considered similar.

I hope this helps clarify the concept for you. If you are still struggling, I suggest seeking additional resources or reaching out to a tutor for further assistance. Good luck!