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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Updated on December 04, 2022

Comments

  • T C
    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

    • T C
      T C almost 7 years
      Oh sorry, it's actually in my exam
    • Admin
      Admin almost 7 years
      In fact AB is symmetric only if A and B commute.
    • T C
      T C almost 7 years
      Yeah. The difficulty here is that they do not
    • Jean Marie
      Jean Marie almost 7 years
      @Rohan as said for example in item 5 of (math.vanderbilt.edu/sapirmv/msapir/jan22.html).
    • Jean Marie
      Jean Marie almost 7 years
      @Rohan ... but the question of the OP is "is AB similar to a symmetric matrix...
    • T C
      T C almost 7 years
      Can people show me a solution and then downvote me later ?
    • πr8
      πr8 almost 7 years
      @chítrungchâu What have you tried?
    • Jean Marie
      Jean Marie almost 7 years