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How can I prove that for all $a \in \mathbb{R}$ the following matrix is diagonalizable? I computed the characteristic polynomial, but I couldn't decompose it into linear factors. This is the polynomial: $t^2-3t+2-a^2$ What should I do next?

Here is the matrix: \begin{bmatrix} 2 & a \\ a & 1 \end{bmatrix}

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  • $\begingroup$ Well, what do you usually do in that case? $\endgroup$
    – Mariano Suárez-Álvarez
    Sep 22, 2016 at 19:57
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    $\begingroup$ I cry. I try to compute the eigenspace but that is not helping me much atm $\endgroup$
    – user355196
    Sep 22, 2016 at 20:01
  • $\begingroup$ You have a symmetric matrix. Does that mean anything? $\endgroup$
    – Doug M
    Sep 22, 2016 at 20:03
  • $\begingroup$ What happens for $a=\sqrt 2$? $\endgroup$
    – N74
    Sep 22, 2016 at 20:18

2 Answers 2

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Find the zeors of the polynomial. If there are two of them, you are done else you have to take a closer look at the eigenspace and check if it is two-dimensional.

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  • $\begingroup$ But how do I found the zeros of this polynomial: t^2-3t+2-a^2=0. i found for a=0 that t=1 and 2. but i need to show that for all a this is okay. And now i showed it for a=0 $\endgroup$
    – user355196
    Sep 22, 2016 at 19:59
  • $\begingroup$ @esma2708, you surely know a formula to find the roots of a degree two polynomial! $\endgroup$
    – Mariano Suárez-Álvarez
    Sep 22, 2016 at 20:00
  • $\begingroup$ Holy moly... the abc-formula haahha thaaank you $\endgroup$
    – user355196
    Sep 22, 2016 at 20:02
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The discriminant of the polynomial is $$ 3^2-4(2-a^2)=1+4a^2 $$ For all real values of $a$, the discriminant is positive. What does this tell you about the roots of the polynomial?

What is a sufficient criterion for diagonalizability? (In other words, you don't need to compute the eigenspaces, in this case.)

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  • $\begingroup$ I found 2 values of t. So it is diagonalizable, because we have two eigenvalues? $\endgroup$
    – user355196
    Sep 22, 2016 at 20:22
  • $\begingroup$ @esma2708 Yes, two distinct real eigenvalues make a $2\times 2$ matrix diagonalizable. $\endgroup$
    – egreg
    Sep 22, 2016 at 21:02

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