Triangularity of a matrix over the unitarian space

[estro]

In my book I have a theorem that every matrix is similar to a triangular matrix if the characteristic polynomial could be wrote as a multiplication of linear terms.
Now suppose we are dealing with a matrix over a unitarian space we know that we can write its characteristic polynomial as a multiplication of linear forms, now what I'm trying to prove is that I can find an orthogonal/orthonormal basis in which the matrix is triangular.

Is this true?
Can you give me a hint how to prove such as thing?
[If this is true many examples that I see in my book could be proved in more compact and neat form, so I think this is very useful theorem...]

[EDIT] I have an idea now to use the fact that every matrix in the unitarian space similar to a jordan matrix, so I only need to show that jordan basis could be transformed into orthonormal jordan basis...

 

[micromass]

Hi estro!

In my book I have a theorem that every matrix is similar to a triangular matrix if the characteristic polynomial could be wrote as a multiplication of linear terms.
Now suppose we are dealing with a matrix over a unitarian space we know that we can write its characteristic polynomial as a multiplication of linear forms, now what I'm trying to prove is that I can find an orthogonal/orthonormal basis in which the matrix is triangular.

Is this true?
Can you give me a hint how to prove such as thing?
[If this is true many examples that I see in my book could be proved in more compact and neat form, so I think this is very useful theorem...]

[EDIT] I have an idea now to use the fact that every matrix in the unitarian space similar to a jordan matrix, so I only need to show that jordan basis could be transformed into orthonormal jordan basis...

Firstly, what is the "unitarian space", I have never heard of this term before...

Secondly, (depending on what unitarian space is), I don't think your theorem is true. If there were an orthogonal basis in which the matrix is triangular, then this would mean that there is a orthogonal basis of (generalized) eigenvectors. But this is not true in general.

 

[estro]

unitarian space is a liner space V over the field C where the Inner product defined.
I'll think about what you said.

 

[micromass]

unitarian space is a liner space V over the field C where the Inner product defined.

Ah, that would be a unitary space. I did see that term once, I completely forgot it