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Let $A$ be an $n$-th order square matrix with complex entries. Which of the following statements are true?

(a) $A$ is always similar to a diagonal matrix.

(b) $A$ is always similar to an upper-triangular matrix.

(c) $A$ is similar to a block diagonal matrix, with each diagonal block of size strictly less than $n$, provided $A$ has at least $2$ distinct eigenvalues.

There are so many matrices which are not diagonalizable like nonzero nilpotent matrices. So (a) is false. I think (b) is true as when we find the rank of a matrix it is converted to a upper-triangular matrix. I have no idea about (c).

Is my thinking correct?

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Your thinking is correct for (a). For (b), the answer you gave is correct, but not the argument. When computing the rank, you find a matrix equivalent to $A$ (that is some $SAT$, $S,T$ invertible), not similar (that is some $T^{-1}AT$, $T$ invertible). For (b) and (c) think of the Jordan normal form. –  martini Sep 18 '12 at 11:27
    
look for Jordan normal form –  chaohuang Sep 18 '12 at 13:29
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1 Answer

For (c), do you know about generalized eigenspaces?

In general, it is hard to give an answer without knowing what you know. There is a theorem saying $A$ is similar to an upper triangular matrix if and only if the characteristic polynomial of $A$ splits into linear factors; if you know about that, then you can answer (b) easily. And even if you don't know that theorem, there may be some other thing you do know from which it can be deduced.

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