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Let $A$ be an $n \times n$ matrix with complex entries. Pick out the true statements.

  1. $A$ is always similar to an upper-triangular matrix.
  2. $A$ is always similar to a diagonal matrix.
  3. $A$ 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.

I have solved $1$ & $2$. $1$ is true and $2$ is false.

But I cannot find how to solve $3$. Can somebody help?

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2 Answers 2

This is true and the a way to see this is to consider the Jordan normal form .

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Hint: Consider the Jordan Normal Form.

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