Is an Upper Triangular Matrix with Equal Diagonal Entries Diagonalizable?

[Treadstone 71]

"Let A be an upper triangular matrix with entires in a field F. Suppose that all the diagonal entries of A are equal. Show that A is diagonalizable if and only if it is diagonal."

I'm reviewing old assignments for a midterm. I remember doing this (backward direction is trivial), I can't remember how, but I remember it was easy. Any hints?

 

[shmoe]

If A is similar to a diagonal matrix D, what can you say about the diagonal entries of D?

 

[Treadstone 71]

What do you mean by similar? Do you mean they differ by a few entries? In that case the diagonal entries of D will be the same of that of A.

 

[shmoe]

A is similar to D if there exists an invertible matrix X where X*A*X^{-1}=D.

Saying "A is similar to a diagonal matrix D" is the same thing as saying "A is diagonalizable" except I find it gives a less cumbersome way to give this diagonal matrix a name.

So what can you say about the entries of this D?

 

[Treadstone 71]

I don't know, but I found a way of doing it:

A is upper triangular, and the diagonal entries are all equal, therefore A has a single eingenvalue e. Suppose A is diagonalizable, then V=null(A-eI), therefore A=eI.