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Im not sure about a question, and need your help.

Is a complex matrix with a single Eigenvalue necessarily diagonalizable?

I'm thinking that the answer is true because the opposite case does happen : There is a diagonalizable matrix with only one eigenvalue, but I dont know how to prove it.

Any suggestions? Thanks in advance.

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  • $\begingroup$ A matrix is diagonalizable if and only if its number of eigenvectors is the same as its dimension. So I this the answer is no. And you can regard real matrix as complex matrix. $\endgroup$
    – eccstartup
    Jul 23, 2013 at 7:11
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    $\begingroup$ The answer is no indeed, since $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ only has the eigenvalue $1$, but is clearly not similar to $I_2$. $\endgroup$
    – zuggg
    Jul 23, 2013 at 7:13

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The matrix $$ \left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right) $$

is real, thus complex, has a single eigenvalue, and is not diagonalizable.

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