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If $k$, $l$ and $n$ are integers such that $1 \leq k \leq l \leq n$, show that there exists an $n \times n$ matrix $A$ and an eigenvalue $\alpha$ of $A$ such that $k$ and $l$ are the geometric and algebraic multiplicities of $\alpha$ with respect to $A$.

I figured out that the following matrix formation an algebraic multiplicity $l$ for $\alpha$
\begin{bmatrix} \lambda_1 & 0 & \dots & 0& 0&\dots &0 \\ 0&\lambda_2 & \dots & 0& 0&\dots &0 \\ .\\ 0 & 0 &\dots &\lambda_{n-k} & 0&\dots &0 \\ 0 & 0 &\dots &0 & \alpha&\dots &0 \\ .\\ 0 & 0 &\dots &0 & 0&\dots &\alpha \\ \end{bmatrix} but I need help regarding the geometric multiplicity.

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  • $\begingroup$ Geometric multiplicity is the dimension of eigenspace generated $\endgroup$ – Alessandro Blasetti Nov 7 '16 at 8:42
  • $\begingroup$ yes. but i cant figure out how to show that such a matrix exists. can you give me some hints? $\endgroup$ – user666 Nov 7 '16 at 9:16
  • $\begingroup$ You can try to calculate explicitly $\ker (\alpha I - A)$ $\endgroup$ – Alessandro Blasetti Nov 7 '16 at 9:28

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