Efficiency in matrix multiplication with diagonalizable matrix. [closed]












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Are there efficient algorithms to compute Ax in the case of A being diagonalizable instead of not being?






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closed as off-topic by José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo Jan 17 at 9:49


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    -1












    $begingroup$


    Are there efficient algorithms to compute Ax in the case of A being diagonalizable instead of not being?






    numerical-linear-algebra







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    closed as off-topic by José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo Jan 17 at 9:49


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo

    If this question can be reworded to fit the rules in the help center, please edit the question.



















      -1












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      -1





      $begingroup$


      Are there efficient algorithms to compute Ax in the case of A being diagonalizable instead of not being?






      numerical-linear-algebra







      share|cite|improve this question









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      Are there efficient algorithms to compute Ax in the case of A being diagonalizable instead of not being?






      numerical-linear-algebra




      numerical-linear-algebra






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      asked Jan 17 at 6:56









      Santiago AranguriSantiago Aranguri

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      closed as off-topic by José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo Jan 17 at 9:49


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo Jan 17 at 9:49


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – José Carlos Santos, Shailesh, Lord_Farin, Henrik, Cesareo

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
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          It's just not going to matter. A randomly chosen complex matrix is diagonalizable with probability $1$ - and finding that diagonalization is considerably more involved than just doing the multiplication. Sure, multiplying by a diagonal matrix is nice and simple, but if we didn't start there, converting to (diagonal * something) form requires two matrix multiplications - one by the similarity matrix, one by its inverse. Two matrix multiplications in order to compute one - not worth it.






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          • $begingroup$
            I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
            $endgroup$
            – nicomezi
            Jan 17 at 10:15


















          1 Answer
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          active

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          It's just not going to matter. A randomly chosen complex matrix is diagonalizable with probability $1$ - and finding that diagonalization is considerably more involved than just doing the multiplication. Sure, multiplying by a diagonal matrix is nice and simple, but if we didn't start there, converting to (diagonal * something) form requires two matrix multiplications - one by the similarity matrix, one by its inverse. Two matrix multiplications in order to compute one - not worth it.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
            $endgroup$
            – nicomezi
            Jan 17 at 10:15
















          1












          $begingroup$

          It's just not going to matter. A randomly chosen complex matrix is diagonalizable with probability $1$ - and finding that diagonalization is considerably more involved than just doing the multiplication. Sure, multiplying by a diagonal matrix is nice and simple, but if we didn't start there, converting to (diagonal * something) form requires two matrix multiplications - one by the similarity matrix, one by its inverse. Two matrix multiplications in order to compute one - not worth it.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
            $endgroup$
            – nicomezi
            Jan 17 at 10:15














          1












          1








          1





          $begingroup$

          It's just not going to matter. A randomly chosen complex matrix is diagonalizable with probability $1$ - and finding that diagonalization is considerably more involved than just doing the multiplication. Sure, multiplying by a diagonal matrix is nice and simple, but if we didn't start there, converting to (diagonal * something) form requires two matrix multiplications - one by the similarity matrix, one by its inverse. Two matrix multiplications in order to compute one - not worth it.






          share|cite|improve this answer









          $endgroup$



          It's just not going to matter. A randomly chosen complex matrix is diagonalizable with probability $1$ - and finding that diagonalization is considerably more involved than just doing the multiplication. Sure, multiplying by a diagonal matrix is nice and simple, but if we didn't start there, converting to (diagonal * something) form requires two matrix multiplications - one by the similarity matrix, one by its inverse. Two matrix multiplications in order to compute one - not worth it.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Jan 17 at 7:31









          jmerryjmerry

          10.1k1225




          10.1k1225












          • $begingroup$
            I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
            $endgroup$
            – nicomezi
            Jan 17 at 10:15


















          • $begingroup$
            I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
            $endgroup$
            – nicomezi
            Jan 17 at 10:15
















          $begingroup$
          I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
          $endgroup$
          – nicomezi
          Jan 17 at 10:15




          $begingroup$
          I think you are confusing two things here. He did not ask for the efficiency of diagonalizing a matrix in order to perform a mutliplication, but if the fact that a matrix is diagonalizable can lead to an algorithm simplifying the multiplication.
          $endgroup$
          – nicomezi
          Jan 17 at 10:15



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