Eigenvalues of a matrix product

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it is a well know result that for two $ntimes n$ matrices $A$ and $B$ the eigenvalues of the product $AB$ are equal to those of the product $BA$.



Are there similar results for a product of $m>2$ matrices? E.g. does the product of four $ntimes n$ matrices $ABCD$ have the same eigenvalues as $ACBD$?



Thanks for your help!










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    The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
    – Qiaochu Yuan
    yesterday















up vote
6
down vote

favorite
1












it is a well know result that for two $ntimes n$ matrices $A$ and $B$ the eigenvalues of the product $AB$ are equal to those of the product $BA$.



Are there similar results for a product of $m>2$ matrices? E.g. does the product of four $ntimes n$ matrices $ABCD$ have the same eigenvalues as $ACBD$?



Thanks for your help!










share|cite|improve this question







New contributor




maxmilgram is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.















  • 2




    The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
    – Qiaochu Yuan
    yesterday













up vote
6
down vote

favorite
1









up vote
6
down vote

favorite
1






1





it is a well know result that for two $ntimes n$ matrices $A$ and $B$ the eigenvalues of the product $AB$ are equal to those of the product $BA$.



Are there similar results for a product of $m>2$ matrices? E.g. does the product of four $ntimes n$ matrices $ABCD$ have the same eigenvalues as $ACBD$?



Thanks for your help!










share|cite|improve this question







New contributor




maxmilgram is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











it is a well know result that for two $ntimes n$ matrices $A$ and $B$ the eigenvalues of the product $AB$ are equal to those of the product $BA$.



Are there similar results for a product of $m>2$ matrices? E.g. does the product of four $ntimes n$ matrices $ABCD$ have the same eigenvalues as $ACBD$?



Thanks for your help!







linear-algebra matrices eigenvalues-eigenvectors






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  • 2




    The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
    – Qiaochu Yuan
    yesterday













  • 2




    The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
    – Qiaochu Yuan
    yesterday








2




2




The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
– Qiaochu Yuan
yesterday





The only thing that can be said is what can be deduced from the first result, which is that eigenvalues are invariant under cyclic permutations, so $ABCD$ has the same eigenvalues as $DABC, CDAB$, and $BCDA$, and that's it.
– Qiaochu Yuan
yesterday











2 Answers
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up vote
3
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accepted










Let check this on orthogonal matrices (e.g. rotations dim. $3 times 3$).



We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.






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    up vote
    2
    down vote













    In fact, $AB$ and $BA$ have the same characteristic polynomial.



    The problem with expecting switches in between various other matrices to preserve eigenvalues, is that the characteristic polynomial does not behave well with multiplication of matrices from left/right.



    In other words, if $A$ and $B$ have the same characteristic polynomial, then this may not be true of $CA$ and $CB$, where $C$ is some other matrix. This is because the trace(sum of eigenvalues) is not multiplicative in general, and we can use this to produce counterexamples.



    To give an example, take $A = beginbmatrix1 0 \ 0 2 endbmatrix$ and $B = beginbmatrix 1 1 \ 0 2endbmatrix$. These matrices have the same characteristic polynomial. For yourself, find a $C$ such that $CA$ and $CB$ have different trace, and therefore different eigenvalues. Find a $D$ such that $AD$ and $BD$ have different trace. It follows that $CA,CB$ and $AD,BD$ don't have the same characteristic polynomials.



    The stumbling block in trying to prove, say , that $CBA$ and $CAB$ have the same eigenvalues, is the inability to assure that multiplying by $C$, after knowing that the eigenvalues of $AB$ and $BA$ are the same, retains the equality of eigenvalues. This is why you can use the $C,A,B,D$ above to find counterexamples of the assertions you were making.



    However, cyclic permutations retain eigenvalues, because we are just applying the "two-matrix" case to deduce this. For example, $ABCD = (AB)(CD)$ has the same eigenvalues as $(CD)(AB) = CDAB$ and so on. So while this is a generalization, it may not feel satisfactory.






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      2 Answers
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      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      3
      down vote



      accepted










      Let check this on orthogonal matrices (e.g. rotations dim. $3 times 3$).



      We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.






      share|cite|improve this answer
























        up vote
        3
        down vote



        accepted










        Let check this on orthogonal matrices (e.g. rotations dim. $3 times 3$).



        We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.






        share|cite|improve this answer






















          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          Let check this on orthogonal matrices (e.g. rotations dim. $3 times 3$).



          We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.






          share|cite|improve this answer












          Let check this on orthogonal matrices (e.g. rotations dim. $3 times 3$).



          We have for example $R_1R_1^TR_2R_2^T=I$ with $1$ as eigenvalues. If we change the order of rotations for example $R_1R_2(R_2R_1)^T=R_1R_2R_1^TR_2^T$ certainly if $R_1$ and $R_2$ are not commutative we would not obtain identity matrix and consequently the eigenvalues are different.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          Widawensen

          4,34121444




          4,34121444




















              up vote
              2
              down vote













              In fact, $AB$ and $BA$ have the same characteristic polynomial.



              The problem with expecting switches in between various other matrices to preserve eigenvalues, is that the characteristic polynomial does not behave well with multiplication of matrices from left/right.



              In other words, if $A$ and $B$ have the same characteristic polynomial, then this may not be true of $CA$ and $CB$, where $C$ is some other matrix. This is because the trace(sum of eigenvalues) is not multiplicative in general, and we can use this to produce counterexamples.



              To give an example, take $A = beginbmatrix1 0 \ 0 2 endbmatrix$ and $B = beginbmatrix 1 1 \ 0 2endbmatrix$. These matrices have the same characteristic polynomial. For yourself, find a $C$ such that $CA$ and $CB$ have different trace, and therefore different eigenvalues. Find a $D$ such that $AD$ and $BD$ have different trace. It follows that $CA,CB$ and $AD,BD$ don't have the same characteristic polynomials.



              The stumbling block in trying to prove, say , that $CBA$ and $CAB$ have the same eigenvalues, is the inability to assure that multiplying by $C$, after knowing that the eigenvalues of $AB$ and $BA$ are the same, retains the equality of eigenvalues. This is why you can use the $C,A,B,D$ above to find counterexamples of the assertions you were making.



              However, cyclic permutations retain eigenvalues, because we are just applying the "two-matrix" case to deduce this. For example, $ABCD = (AB)(CD)$ has the same eigenvalues as $(CD)(AB) = CDAB$ and so on. So while this is a generalization, it may not feel satisfactory.






              share|cite|improve this answer
























                up vote
                2
                down vote













                In fact, $AB$ and $BA$ have the same characteristic polynomial.



                The problem with expecting switches in between various other matrices to preserve eigenvalues, is that the characteristic polynomial does not behave well with multiplication of matrices from left/right.



                In other words, if $A$ and $B$ have the same characteristic polynomial, then this may not be true of $CA$ and $CB$, where $C$ is some other matrix. This is because the trace(sum of eigenvalues) is not multiplicative in general, and we can use this to produce counterexamples.



                To give an example, take $A = beginbmatrix1 0 \ 0 2 endbmatrix$ and $B = beginbmatrix 1 1 \ 0 2endbmatrix$. These matrices have the same characteristic polynomial. For yourself, find a $C$ such that $CA$ and $CB$ have different trace, and therefore different eigenvalues. Find a $D$ such that $AD$ and $BD$ have different trace. It follows that $CA,CB$ and $AD,BD$ don't have the same characteristic polynomials.



                The stumbling block in trying to prove, say , that $CBA$ and $CAB$ have the same eigenvalues, is the inability to assure that multiplying by $C$, after knowing that the eigenvalues of $AB$ and $BA$ are the same, retains the equality of eigenvalues. This is why you can use the $C,A,B,D$ above to find counterexamples of the assertions you were making.



                However, cyclic permutations retain eigenvalues, because we are just applying the "two-matrix" case to deduce this. For example, $ABCD = (AB)(CD)$ has the same eigenvalues as $(CD)(AB) = CDAB$ and so on. So while this is a generalization, it may not feel satisfactory.






                share|cite|improve this answer






















                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  In fact, $AB$ and $BA$ have the same characteristic polynomial.



                  The problem with expecting switches in between various other matrices to preserve eigenvalues, is that the characteristic polynomial does not behave well with multiplication of matrices from left/right.



                  In other words, if $A$ and $B$ have the same characteristic polynomial, then this may not be true of $CA$ and $CB$, where $C$ is some other matrix. This is because the trace(sum of eigenvalues) is not multiplicative in general, and we can use this to produce counterexamples.



                  To give an example, take $A = beginbmatrix1 0 \ 0 2 endbmatrix$ and $B = beginbmatrix 1 1 \ 0 2endbmatrix$. These matrices have the same characteristic polynomial. For yourself, find a $C$ such that $CA$ and $CB$ have different trace, and therefore different eigenvalues. Find a $D$ such that $AD$ and $BD$ have different trace. It follows that $CA,CB$ and $AD,BD$ don't have the same characteristic polynomials.



                  The stumbling block in trying to prove, say , that $CBA$ and $CAB$ have the same eigenvalues, is the inability to assure that multiplying by $C$, after knowing that the eigenvalues of $AB$ and $BA$ are the same, retains the equality of eigenvalues. This is why you can use the $C,A,B,D$ above to find counterexamples of the assertions you were making.



                  However, cyclic permutations retain eigenvalues, because we are just applying the "two-matrix" case to deduce this. For example, $ABCD = (AB)(CD)$ has the same eigenvalues as $(CD)(AB) = CDAB$ and so on. So while this is a generalization, it may not feel satisfactory.






                  share|cite|improve this answer












                  In fact, $AB$ and $BA$ have the same characteristic polynomial.



                  The problem with expecting switches in between various other matrices to preserve eigenvalues, is that the characteristic polynomial does not behave well with multiplication of matrices from left/right.



                  In other words, if $A$ and $B$ have the same characteristic polynomial, then this may not be true of $CA$ and $CB$, where $C$ is some other matrix. This is because the trace(sum of eigenvalues) is not multiplicative in general, and we can use this to produce counterexamples.



                  To give an example, take $A = beginbmatrix1 0 \ 0 2 endbmatrix$ and $B = beginbmatrix 1 1 \ 0 2endbmatrix$. These matrices have the same characteristic polynomial. For yourself, find a $C$ such that $CA$ and $CB$ have different trace, and therefore different eigenvalues. Find a $D$ such that $AD$ and $BD$ have different trace. It follows that $CA,CB$ and $AD,BD$ don't have the same characteristic polynomials.



                  The stumbling block in trying to prove, say , that $CBA$ and $CAB$ have the same eigenvalues, is the inability to assure that multiplying by $C$, after knowing that the eigenvalues of $AB$ and $BA$ are the same, retains the equality of eigenvalues. This is why you can use the $C,A,B,D$ above to find counterexamples of the assertions you were making.



                  However, cyclic permutations retain eigenvalues, because we are just applying the "two-matrix" case to deduce this. For example, $ABCD = (AB)(CD)$ has the same eigenvalues as $(CD)(AB) = CDAB$ and so on. So while this is a generalization, it may not feel satisfactory.







                  share|cite|improve this answer












                  share|cite|improve this answer



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                  answered yesterday









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