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.