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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...

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...

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