If $p$ & $q$ are eigenvalues of $A$ & $B$ respectively, then are $p+q$ & $pq$ that of $A+B$ & $AB$ respectively?

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I really can't get it, either a proof or a way to disprove. Suppose $AB=BA$? Does this result hold then? Here, $A$ & $B$ are any $ntimes n$ matrices.



If so then how? Please help.










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    up vote
    0
    down vote

    favorite












    I really can't get it, either a proof or a way to disprove. Suppose $AB=BA$? Does this result hold then? Here, $A$ & $B$ are any $ntimes n$ matrices.



    If so then how? Please help.










    share|cite|improve this question

























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I really can't get it, either a proof or a way to disprove. Suppose $AB=BA$? Does this result hold then? Here, $A$ & $B$ are any $ntimes n$ matrices.



      If so then how? Please help.










      share|cite|improve this question















      I really can't get it, either a proof or a way to disprove. Suppose $AB=BA$? Does this result hold then? Here, $A$ & $B$ are any $ntimes n$ matrices.



      If so then how? Please help.







      linear-algebra






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      edited Sep 8 at 15:22









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      asked Sep 8 at 3:21









      Priya Dey

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          For $A+B$, say $A=beginpmatrix2&0\0&0endpmatrix$ and $B=beginpmatrix0&0\0&1endpmatrix$. Is $3$ an e-value of $A+B$?



          Secondly, using the same $A$ and $B$, say. $AB=0$. Hence $2cdot1$ is not an e-value...






          share|cite|improve this answer




















          • is there any condition under which the above is true?
            – Priya Dey
            Sep 8 at 6:48










          • I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
            – Chris Custer
            Sep 8 at 6:58










          • If $p$ and $q$ were e-values with a common e-vector, say.
            – Chris Custer
            Sep 8 at 7:15











          • this means commutativity of A &B also fails to have the above property?
            – Priya Dey
            Sep 8 at 7:24










          • It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
            – Chris Custer
            Sep 8 at 7:40










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          up vote
          2
          down vote













          For $A+B$, say $A=beginpmatrix2&0\0&0endpmatrix$ and $B=beginpmatrix0&0\0&1endpmatrix$. Is $3$ an e-value of $A+B$?



          Secondly, using the same $A$ and $B$, say. $AB=0$. Hence $2cdot1$ is not an e-value...






          share|cite|improve this answer




















          • is there any condition under which the above is true?
            – Priya Dey
            Sep 8 at 6:48










          • I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
            – Chris Custer
            Sep 8 at 6:58










          • If $p$ and $q$ were e-values with a common e-vector, say.
            – Chris Custer
            Sep 8 at 7:15











          • this means commutativity of A &B also fails to have the above property?
            – Priya Dey
            Sep 8 at 7:24










          • It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
            – Chris Custer
            Sep 8 at 7:40














          up vote
          2
          down vote













          For $A+B$, say $A=beginpmatrix2&0\0&0endpmatrix$ and $B=beginpmatrix0&0\0&1endpmatrix$. Is $3$ an e-value of $A+B$?



          Secondly, using the same $A$ and $B$, say. $AB=0$. Hence $2cdot1$ is not an e-value...






          share|cite|improve this answer




















          • is there any condition under which the above is true?
            – Priya Dey
            Sep 8 at 6:48










          • I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
            – Chris Custer
            Sep 8 at 6:58










          • If $p$ and $q$ were e-values with a common e-vector, say.
            – Chris Custer
            Sep 8 at 7:15











          • this means commutativity of A &B also fails to have the above property?
            – Priya Dey
            Sep 8 at 7:24










          • It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
            – Chris Custer
            Sep 8 at 7:40












          up vote
          2
          down vote










          up vote
          2
          down vote









          For $A+B$, say $A=beginpmatrix2&0\0&0endpmatrix$ and $B=beginpmatrix0&0\0&1endpmatrix$. Is $3$ an e-value of $A+B$?



          Secondly, using the same $A$ and $B$, say. $AB=0$. Hence $2cdot1$ is not an e-value...






          share|cite|improve this answer












          For $A+B$, say $A=beginpmatrix2&0\0&0endpmatrix$ and $B=beginpmatrix0&0\0&1endpmatrix$. Is $3$ an e-value of $A+B$?



          Secondly, using the same $A$ and $B$, say. $AB=0$. Hence $2cdot1$ is not an e-value...







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Sep 8 at 3:42









          Chris Custer

          6,6192622




          6,6192622











          • is there any condition under which the above is true?
            – Priya Dey
            Sep 8 at 6:48










          • I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
            – Chris Custer
            Sep 8 at 6:58










          • If $p$ and $q$ were e-values with a common e-vector, say.
            – Chris Custer
            Sep 8 at 7:15











          • this means commutativity of A &B also fails to have the above property?
            – Priya Dey
            Sep 8 at 7:24










          • It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
            – Chris Custer
            Sep 8 at 7:40
















          • is there any condition under which the above is true?
            – Priya Dey
            Sep 8 at 6:48










          • I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
            – Chris Custer
            Sep 8 at 6:58










          • If $p$ and $q$ were e-values with a common e-vector, say.
            – Chris Custer
            Sep 8 at 7:15











          • this means commutativity of A &B also fails to have the above property?
            – Priya Dey
            Sep 8 at 7:24










          • It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
            – Chris Custer
            Sep 8 at 7:40















          is there any condition under which the above is true?
          – Priya Dey
          Sep 8 at 6:48




          is there any condition under which the above is true?
          – Priya Dey
          Sep 8 at 6:48












          I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
          – Chris Custer
          Sep 8 at 6:58




          I guess if $A$ and $B$ were the identity matrix, off the top of my head... maybe even multiples of the identity. I'd guess there might be other examples (where it's true).
          – Chris Custer
          Sep 8 at 6:58












          If $p$ and $q$ were e-values with a common e-vector, say.
          – Chris Custer
          Sep 8 at 7:15





          If $p$ and $q$ were e-values with a common e-vector, say.
          – Chris Custer
          Sep 8 at 7:15













          this means commutativity of A &B also fails to have the above property?
          – Priya Dey
          Sep 8 at 7:24




          this means commutativity of A &B also fails to have the above property?
          – Priya Dey
          Sep 8 at 7:24












          It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
          – Chris Custer
          Sep 8 at 7:40




          It won't always have it. Notice in my example above $A$ and $B$ commute. In fact $AB=BA=0$.
          – Chris Custer
          Sep 8 at 7:40

















           

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