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Using Jordan standard form, we can prove

Any complex matrix can be decomposed into the product of two symmetric matrices, and one of them is invertible.

then how to prove that

Any real square matrix can be decomposed into the product of two symmetric matrices, and at least one of them is invertible. ?

Thanks for your help.

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  • 1
    $\begingroup$ see this math.stackexchange.com/questions/3000326/… $\endgroup$ – Widawensen Nov 28 '18 at 13:58
  • $\begingroup$ If I remember correctly (but I'm not 100% sure), Irving Kaplansky's Linear Algebra and Geometry: A Second Course (Dover) also has a proof. $\endgroup$ – user1551 Nov 28 '18 at 14:05
  • $\begingroup$ @Widawensen thanks. $\endgroup$ – Fyhswdsxjj Nov 28 '18 at 15:05
  • $\begingroup$ @user1551 thanks. $\endgroup$ – Fyhswdsxjj Nov 28 '18 at 15:05

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