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a) Unitary matrices are normal matrix hence diagonalizable as a consequence of spectral theorem

b)same as a)

c)No idea.but I think it may not be diagonalizable unless it has one eigen value with dimension of eigen space $1$

d) No idea.

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Hint: The shear matrix $$\left[\begin{array}{cc} 1 & 1\\ 0 & 1 \end{array}\right]$$ has two real eigenvalues (both equal $1$) complex entries, and cannot be diagonalized.

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    thanku, could you just confirm me am I right in a,b,c?2012-12-18
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    @Kuttus: $a$ and $b$ are correct.2012-12-18
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    what is strictly upper triangular matrix?2012-12-18
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    @Kuttus: Not sure. I assumed it just meant upper triangular.2012-12-18
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    do you know the answer of $c$? I am thinking still though2012-12-18
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    @Kuttus A matrix is called *strictly upper triangular* if it is upper triangular and its main diagonal is zero.2012-12-19