# Upper Hessenberg zero-column sum matrix

1. May 2, 2012

### mansolo

Let A be an (n x n) upper Hessenberg matrix with the following constraints:
a_{j,j} < 0 , j=1,...,n
a_{j,j}= - sum a_{i,j} [i≠j] , j=1,...,n-1
a_{n,n} > - sum a_{i,n} [i≠n]
Example:

-1 2 0 2 0
1 -5 4 0 0
0 3 -9 1 0
0 0 5 -5 3
0 0 0 2 -7

I want to prove A is diagonalizable (I am pretty sure it is, but haven't
found a formal demostration)

Hint: If we make a_{1,4}=0; a_{4,4}=-3, then it is a tridiagonal matrix
similar to a Hermitian one, and therefore diagonalizable.

_M.

2. May 2, 2012

### DonAntonio

I know what is the Heisenberg group of upper 3 x 3 matrices, I know what is an upper matrix and I even

have some idea what is Heisenberg's Matrix mechanics...

Your matrix matrix is neither 3 x 3, nor upper and, as far as I know, now Heisenberg's as, even if we'd talk of higher dimensions

Heisenberg groups, we need upper matrices, so: what exactly is your matrix, anyway??

Anyway, using this nice site http://calculator-online.org/s/matrix/sobstvennyie/ it seems to be your matrix has

5 different complex eigenvalues so it is diagonalizable over the complex field.

DonAntonio