Product of two symmetric matrices is similar to a symmetric matrix
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Here is a counter-example of two symmetric matrices $A$, $B$ whose product, besides being non symmetrical, cannot be similar to a symmetric matrix.
Consider matrices
$$A=\pmatrix{1&2\\2&1} \ \ \ \text{and} \ \ \ B=\pmatrix{1&0\\0&-1}.$$
$AB=\pmatrix{1&-2\\2&-1}.$ which is non symmetric.
Moreover, the characteristic polynomial of $AB$ is $\lambda^2+3$: thus, the eigenvalues of $AB$ are $\pm i \sqrt{3}$. If it was similar to a symmetric matrix, it would have the same real eigenvalues.
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T C
Updated on December 04, 2022Comments
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T C 11 months
Let $A,B$ be symmetric real matrices. Is $AB$ similar to a symmetric matrix?
This is a problem in my exam. Not a conjecture :v
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T C almost 7 yearsOh sorry, it's actually in my exam
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Admin almost 7 yearsIn fact AB is symmetric only if A and B commute.
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T C almost 7 yearsYeah. The difficulty here is that they do not
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Jean Marie almost 7 years@Rohan as said for example in item 5 of (math.vanderbilt.edu/sapirmv/msapir/jan22.html).
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Jean Marie almost 7 years@Rohan ... but the question of the OP is "is AB similar to a symmetric matrix...
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T C almost 7 yearsCan people show me a solution and then downvote me later ?
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πr8 almost 7 years@chítrungchâu What have you tried?
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Jean Marie almost 7 yearsSomewhat related (mathoverflow.net/q/106191)(http://math.stackexchange.com/q/….
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