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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.2012-09-18
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    look for Jordan normal form2012-09-18

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