Eigenvalues and Eigenvectors of diagonal marix












0












$begingroup$


Problem:




Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.



Find $sigma$($D$) and all eigenvectors of $D$.




My thoughts:



As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .



But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.



Can someone help me?






linear-algebra eigenvalues-eigenvectors







share|cite|improve this question











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    0












    $begingroup$


    Problem:




    Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.



    Find $sigma$($D$) and all eigenvectors of $D$.




    My thoughts:



    As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .



    But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.



    Can someone help me?






    linear-algebra eigenvalues-eigenvectors







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$


      Problem:




      Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.



      Find $sigma$($D$) and all eigenvectors of $D$.




      My thoughts:



      As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .



      But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.



      Can someone help me?






      linear-algebra eigenvalues-eigenvectors







      share|cite|improve this question











      $endgroup$




      Problem:




      Let $D$:= diag($lambda_1, ldots, lambda_n$), i.e., $D$ is a diagonal matrix in $mathbb{C}^{ntimes n}$ with entries $lambda_1, ldots, lambda_n$ ∈ $mathbb{C}$ on its diagonal.



      Find $sigma$($D$) and all eigenvectors of $D$.




      My thoughts:



      As the spectrum of $D$ is the set of all eigenvalues, then $sigma$($D$) should be just $lambda_1 cdots lambda_n$ = $mathbb {lambda_n}^{n}$ .



      But how can I find the eigenvectors ? I know I have to calculate the $D$ - $lambda I$.



      Can someone help me?






      linear-algebra eigenvalues-eigenvectors




      linear-algebra eigenvalues-eigenvectors






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 6 at 9:31









      Berkheimer

      1,4371024




      1,4371024










      asked Jan 6 at 9:21









      KaiKai

      636




      636






















          1 Answer
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          $begingroup$

          If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:



          $$
          e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
          $$






          share|cite|improve this answer









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            1 Answer
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            active

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            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            3












            $begingroup$

            If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:



            $$
            e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
            $$






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:



              $$
              e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
              $$






              share|cite|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$

                If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:



                $$
                e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
                $$






                share|cite|improve this answer









                $endgroup$



                If the matrix is diagonal, the eigenvectors are just the standard basis of $mathbb{C}$:



                $$
                e_1 = (1, 0, dots, 0)^t, dots, e_n=(0,dots, 0, 1)^t .
                $$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 6 at 9:33









                d.t.d.t.

                14.2k23074




                14.2k23074






























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