Let $p(x) = a_0 + a_1x + · · · + a_nx^n$ be the characteristic polynomial of a
$n × n$ matrix $A$ with entries in $\mathbb{R}$. Then which of the following statements is
true?
(a) $p(x)$ has no repeated roots.
(b) $p(x)$ can be expressed as a product of linear polynomials with real coefficients.
(c) If $p(x)$ can be expressed as a product of linear polynomials with real
coefficients then there is a basis of $\mathbb{R}^n$ consisting of eigenvectors of $A$
(a)obviously $p(x)$ has repeated roots as the identity matrix has all roots as $1$.so it is false
(b)real matrix can have imaginary eigen values.so its false.
(c)Not sure.
am I right for (a) and (b).
Can I get some help for (c).
Best Answer
A simple counter-example to (c) $$ A = \left[\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}\right] $$ The characteristic polynomial is $\lambda^{2}$, which is a product of two linear terms, but $A$ cannot be diagonalized. If it could be diagonalized, then the eigenvalues would have to be 0, which would make the diagonal matrix 0.