Explicit formula for inner product given orthonormal basis












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Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?






linear-algebra complex-numbers algorithms inner-product-space orthonormal







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    $begingroup$


    Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?






    linear-algebra complex-numbers algorithms inner-product-space orthonormal







    share|cite|improve this question









    $endgroup$















      2












      2








      2


      1



      $begingroup$


      Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?






      linear-algebra complex-numbers algorithms inner-product-space orthonormal







      share|cite|improve this question









      $endgroup$




      Suppose $V$ is a $mathbb C$-vector space and $v_1, v_2, ldots v_n in V$ (given explicitly, for instance as vectors with given elements) form its orthonormal basis relative to some inner product $langlecdot{,}cdotrangle$. Is there any algorithm for explicit formula for this inner product?






      linear-algebra complex-numbers algorithms inner-product-space orthonormal




      linear-algebra complex-numbers algorithms inner-product-space orthonormal






      share|cite|improve this question













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      asked Jan 25 at 14:05









      enedilenedil

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          Using the sesquilinearity of the inner product, we have
          $$
          left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
          $$
          That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".






          share|cite|improve this answer









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            $begingroup$

            Using the sesquilinearity of the inner product, we have
            $$
            left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
            $$
            That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".






            share|cite|improve this answer









            $endgroup$


















              2












              $begingroup$

              Using the sesquilinearity of the inner product, we have
              $$
              left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
              $$
              That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".






              share|cite|improve this answer









              $endgroup$
















                2












                2








                2





                $begingroup$

                Using the sesquilinearity of the inner product, we have
                $$
                left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
                $$
                That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".






                share|cite|improve this answer









                $endgroup$



                Using the sesquilinearity of the inner product, we have
                $$
                left langle sum_{j=1}^n alpha_j v_j , sum_{k=1}^n beta_K v_k right rangle = sum_{j=1}^n sum_{k=1}^n alpha_j overline{beta_k} langle v_j,v_k rangle = sum_{j=1}^n alpha_j overline{beta_j}
                $$
                That is: if we express vectors using their coordinates relative to this orthonormal basis, then the inner product can be computed using the usual "dot-product".







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 25 at 14:11









                OmnomnomnomOmnomnomnom

                129k792185




                129k792185






























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