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If a $3 \times 3$ matrix is diagonalizable and has eigenvalues $1$ and $2$ but has two eigenvectors with eigenvalue $2$, would we have the eigenvalue matrix $$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix}?$$

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    Yes, that is one possibility. Of course you can permute the diagonal elements of the diagonal matrix2012-12-03
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    @Stefan ahh, I just wanted to know whether you entered in the value 2 twice for each of the eigenvectors2012-12-03
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    Are the eigenvectors, with eigenvalue 2, linearly independent?2012-12-03
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    Also note that both eigenvectors for the eigenvalue 2 have to be linear independent.2012-12-03
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    They are, but what if that wasn't the case?2012-12-03
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    For a matrix to be diagonalizable, $\mathbb R^3$ has to be the direct sum of the eigenspaces. From this it follows, that if you have 2 eigenvalues, one of the two eigenspaces has to have dimension 2, and thus there have to exist two linear independent eigenvectors to that eigenvalue.2012-12-03

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Since the linear operator described by this matrix acts on 3-dimensional space, and is known to have

  • a one-dimensional eigenspace for eigenvalue $1$, and
  • two-dimensional eigenspace for eigenvalue 2,

it is correct to conclude that the space has a basis composed of eigenvectors, and in this basis the operator has the matrix $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{pmatrix} $$ up to permutation of rows.


If both eigenspaces turned out to be one-dimensional, the Jordan canonical form of this matrix could have been $$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 1 \\ 0 & 0 & 2 \end{pmatrix} $$ or $$ \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{pmatrix} $$