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Let $p(x)=x^3-6x^2+11x-6$. Describe the Jordan Canonical forms of the $2\times 2-$ matrices $M$ that satisfy $p(M)=0$ if any such matrices exist.

$p(x)=(x-1)(x-2)(x-3)$. so the possible eigenvalues are $1,2,3$. (since its minimal polynomial must divide $p(x)$.)

My problem is showing the possible jordan form of matrices having repetitive eigenvalues . For example, if $\lambda=1,1$, the possible jordan forms are

$$\begin{bmatrix} 1&0\\0&1 \end{bmatrix} and \: \begin{bmatrix} 1&1\\0&1 \end{bmatrix},$$

I am not sure if we can accept both of these matrices. Do I have to check these two matrices annihilates the polynomial $p(x)$?

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    The second matrix does not work. Its minimal polynomial is $(x-1)^2$. – 2017-01-23

1 Answers 1

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The minimal polynomial of a matrix divides any polynomial that the matrix satisfies. The minimal polynomial's degree is also less than or equal to the size of the matrix.

Therefore the possible minimal polynomials for such a matrix are:

$$ x-1, \ x-2, \ x-3, \ (x-1)(x-2), \ (x-1)(x-3), \textrm{ or } (x-2)(x-3).$$

The Jordan Form corresponding to $(x-1)$ would be $$\begin{pmatrix}1 & 0 \\ 0 & 1 \end{pmatrix}_.$$

I'll leave it to you to create the other Jordan forms from here.