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I would like to simultaneously diagonalise the quadratic forms

$A=2x^2+3y^2+3z^2-2yz$,

and

$B=x^2+3y^2+3z^2+6xy+2yz-6zx$.

Of course there's a theorem saying this is possible and I followed the proof (diagonalise, via $P$, the positive definite $A$ to $I$ first and then find a unitary matrix to diagonalise $P^TBP$ etc.) but I ended up with really complicated matrices that made me doubt if the question intended me to do it that way. Could you please let me know if there are clever ways to solve this please. Preferably the clever way is also systematic.

Thanks!

dcwang
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  • I have a paperback of Matrix Analysis by Horn and Johnson. Seems to be the same as the original 1985 edition. Section 4.5 in this edition is pages 218-243, is called Congruence and simultaneous diagonalization of Hermitian and symmetric matrices. The best part is the table of results on page 229. For your problem page 231 is specific. I did it, it can be done with nice numbers. Fair amount of work. There seem to be other, very different versions of the book. I get it, the second edition is 2013. – Will Jagy Aug 23 '15 at 02:53
  • Thanks for the reference Will but may I ask what method you used to do this particular problem? Is it any different from what I mentioned in my above post. – dcwang Aug 23 '15 at 04:41
  • http://math.stackexchange.com/questions/88022/congruence-and-diagonalizations – Will Jagy Aug 23 '15 at 17:13
  • see https://books.google.com/books?id=RO_TMc7ETPEC&pg=PA430&lpg=PA430&dq=in+an+appropriate+coordinate+system,+the+form+of+the+strictly+positive+definite+one+becomes&source=bl&ots=8Mo3A8OTX2&sig=eLBSQ3NwWBrWxpZAs3jlSbLxQhQ&hl=en&sa=X&ved=0CB4Q6AEwAGoVChMIxI2f-Oa_xwIVUTeICh3rxQCN#v=onepage&q=in%20an%20appropriate%20coordinate%20system%2C%20the%20form%20of%20the%20strictly%20positive%20definite%20one%20becomes&f=false Where did you get your question? – Will Jagy Aug 23 '15 at 18:22
  • Thanks, it was an exercise sheet question. – dcwang Aug 27 '15 at 09:27

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