TITLE: Is it possible to find a companion matrix of a polynomial which is also hermitian?
QUESTION [7 upvotes]: The eigenvalues of a square matrix $A$ coincide with the roots of its characteristic polynomial $p[A]$. Conversely, if I have a polynomial
$$
a_0 + a_1 x + \cdots + a_{n-1}x^{n-1} + x^n ~,
$$
I can define a companion matrix
$$
A[p]=\begin{bmatrix}
0 & 0 & \dots & 0 & -a_0 \\
1 & 0 & \dots & 0 & -a_1 \\
0 & 1 & \dots & 0 & -a_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \dots & 1 & -a_{n-1}
\end{bmatrix}.$$
The characteristic polynomial of the matrix $A[p]$ is the original polynomial $p$. 
The companion matrix defined in this way is not Hermitian. 
Edit: Consider only hyperbolic polynomials, i.e., polynomials which have only real roots. 
However, the companion matrix is not the only matrix whose characteristic polynomial is the polynomial $p$. For an example, just consider the diagonal matrix of the eigenvalues ${\rm diag}{(\lambda_1,\lambda_2,\ldots,\lambda_n)}$. This matrix is indeed Hermitian, but one needs to calculate eigenvalues, which can be a difficult task for large matrix sizes.
Therefore my question is:
Is it possible to define a "companion" matrix (a matrix whose characteristic polynomial is the given polynomial $p$) which is Hermitian, and which is easier to calculate than the diagonal matrix of the eigenvalues ${\rm diag}{(\lambda_1,\lambda_2,\ldots,\lambda_n)}$? With easy to calculate I mean a matrix which can be written in terms of the parameters $a_i$ and without calculating the eigenvalues.

REPLY [5 votes]: It's a little sad to see the previous discouraging answer accepted relatively quickly. In fact, there is a construction due to Miroslav Fiedler and improved by Gerhard Schmeisser that constructs a tridiagonal matrix whose characteristic polynomial is (up to a constant factor) the input polynomial, by using a modified Euclidean algorithm to effectively generate Sturmian sequences (which was mentioned by Robert Israel in a comment to the OP). If the subdiagonal elements ($c_k$ in the notation of Schmeisser's paper) generated are all positive, then the tridiagonal matrix is indeed symmetric, and thus has its roots all real.
See the papers for more details. Mathematica code implementing the construction can be found here; it should not be too difficult to adapt it to other languages.