If we are given a $3\times 3$ real matrix $A$ that is both symmetric and orthogonal, then what can we say about $A$ based on the Spectral Theorem?
Spectral Theorem
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linear-algebra
1 Answers
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Hint: what real numbers have absolute value $1$?
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0Well, I know that the diagonal entries have to be $\pm 1$, but I was thinking if there is more to it, since it is asking about a $3\times 3$ matrix? – 2012-11-29
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2?? only after diagonalization are the diagonal entries $\rm 1$. – 2012-11-29
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0$\pm 1$, that is. – 2012-11-29
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0@gedgar: yes thats correct...however my trouble is on how to phrase my answer. – 2012-11-29
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0Does it suffice to say something along the lines: since the matrix is symmetric and orthogonal, then the matrix A would be normal, so there is some orthogonal matrix P such that $P^tAP$ is a diagonal matrix whose entries are all 1s. Would this suffice or is it asking for more, sth else?? Thanks – 2012-11-29
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0@ froggie: thanks! Regarding the question the only other thing I can get is that the matrix A will have real eigenvalues. – 2012-11-29
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0@ Robert Israel: so is that all what the problem is asking? Namely to assert that the diagonal entries of the matrix A will be $\pm 1$ after diagonalization? It seems too simple! – 2012-11-29
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0No, we can say more, if we want. This is a "reflection" ... in a point, or in a line, or in a plane (or in the whole space i.e. the identity). – 2012-11-29