1
$\begingroup$

If I am given the characteristic polynomial for some matrix $A$, $p(λ) = (λ − 2)(λ +5)(λ − 1)$, is this enough info to determine if the matrix $A$ is diagonalizable? I read a theorem in a textbook that matrix $A$ is diagonalizable if the dimension of the eigenspace with its corresponding eigenvalue is equal to the the degree of the eigenvalue in the characteristic polynomial (i.e. say the $\dim(E_λ) = 2$ when $λ = 2$ and the degree is $(λ - 2)^1$). However, is there anyway to determine this without using this theorem?

$\endgroup$
2
$\begingroup$

$A$ is diagonalizable $\Leftrightarrow$ minimal polynomial of $A$ splits into linear factors. In your case assuming $A$ is a $3\times 3$ matrix, the eigen values are distinct and have multiplicity 1 and also minimal polynomial is same as characteristic polynomial. Hence its diagonalizable.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.