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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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up vote 1 down vote accepted

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? – Un Chien Andalou Dec 18 '12 at 18:07
@Kuttus: $a$ and $b$ are correct. – Eric Naslund Dec 18 '12 at 18:07
what is strictly upper triangular matrix? – Un Chien Andalou Dec 18 '12 at 18:12
@Kuttus: Not sure. I assumed it just meant upper triangular. – Eric Naslund Dec 18 '12 at 18:14
@Kuttus A matrix is called strictly upper triangular if it is upper triangular and its main diagonal is zero. – user1551 Dec 19 '12 at 4:12

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