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Let $S = \{\lambda_1, \cdots, \lambda_n\}$ be an ordered set of $n$ real numbers, not all equal, but not all necessarily distinct. Pick out the true statements:

a. There exists an $n\times n$ matrix with complex entries, which is not self-adjoint, whose set of eigenvalues is given by $S$.

b. There exists an $n\times n$ self-adjoint, non-diagonal matrix with complex entries whose set of eigenvalues is given by $S$.

c. There exists an $n\times n$ symmetric, non-diagonal matrix with real entries whose set of eigenvalues is given by $S$.

a) No Idea.

b) as hermitian matrices(self adjoint) matrices has real eigen values only so $b$ may be true..

c) same logic as $b$. please help.

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Hint for a): What are eigenvalues of a diagonal matrix? – Hans Giebenrath Dec 16 '12 at 13:35
just diagonal entries.... so?where do I apply that not self adjoint? – Un Chien Andalou Dec 16 '12 at 13:46
Take a "proper" upper triangular matrix. – Hans Giebenrath Dec 16 '12 at 14:06
@Kuttus Just curious. Do your questions in this and the two or three recent posts come from a textbook? If so, would you mind telling me the its title? – user1551 Dec 16 '12 at 15:36
No not a text book but from a national level PhD and Masters Fellowship selection test question papers of past years in India. – Un Chien Andalou Dec 16 '12 at 15:51
up vote 2 down vote accepted

For any given set of real numbers, one can find a hermitian (self-adjoint) matrix with those entries as eigenvalues. One easy way to see this is, stack all your real numbers in the diagonal of a diagonal matrix $D$. Let $U$ be any unitary matrix (means with complex entries). Then $A=UDU^H$ is a hermitian (self-adjoint) matrix with the diagonal entries of D as its eigenvalues.

Now consider any non self-adjoint invertible matrix $P$. Then $B=PAP^{-1}$ is also a complex matrix which is not self-adjoint but has same eigenvalues as $A$ (why?).

Now consider any orthonormal matrix $Q$ (distinguish orthonormal and unitary), then $C=QDQ^T$ is a symmetric matrix (with real entries) with the given real numbers as eigenvalues.

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yes as $A$ and $B$ are similar matrix they have same eigen values, could you explain your last 2 lines? – Un Chien Andalou Dec 16 '12 at 14:07
what do u mean by orthonormal matrix? in usual sense? $v_iv_j=1,0$ as $i= j$ and $i\neq j$ respectively? $v_i$'s are collumn vectors – Un Chien Andalou Dec 16 '12 at 14:24
yes, in the usual sense only, but orthonormal matrix will be containing only real entries whereas unitary matrix will be containing complex entries. – dineshdileep Dec 16 '12 at 16:57
so only $c$ is true – Un Chien Andalou Jun 3 '13 at 18:23
no, all of them are true, please read the answer carefully, situations corresponding to all three situations are explained – dineshdileep Jun 4 '13 at 4:14

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