Characteristic polynomial of square of matrix?
By Emily Wilson •
Let $A \in M_n(\mathbb{C})$ with $p_A(x)=(-1)^n\cdot (x-λ_1)\dots(x-λ_n)$ it's characteristic polynomial.
Prove that $p_{A^2}(x)=(-1)^n\cdot (x-λ_1^2)\dots(x-λ_n^2)$
Any tips on where to start from?
$\endgroup$ 11 Answer
$\begingroup$The characteristic polynomial of a matrix has roots that are equal to its eigenvalues. Let $x$ be an eigenvector of $A$ with corresponding eigenvalue $\lambda$. Then $$ AAx = A \lambda x = \lambda^2 x. $$ So $x$ is an eigenvector of $A^2$ too, but its eigenvalue is $\lambda^2$. So the roots of the characteristic polynomial of $AA$ are the squares of the roots of $A$. Its characteristic polynomial is completely determined by this fact.
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