Does a $2times2$ matrix B exist such that B is diagonalizable but B is not invertible?












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Is it possible? I know that for a matrix to be diagonalizable it needs sufficient linearly independent eigenvectors to make up invertible matrix P, but does that mean B itself must also be invertible?






linear-algebra diagonalization







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    Yes it is. For example take the matrix $[1 0; 0 0]$.
    $endgroup$
    – AnyAD
    Dec 12 '18 at 3:48


















1












$begingroup$


Is it possible? I know that for a matrix to be diagonalizable it needs sufficient linearly independent eigenvectors to make up invertible matrix P, but does that mean B itself must also be invertible?






linear-algebra diagonalization







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Yes it is. For example take the matrix $[1 0; 0 0]$.
    $endgroup$
    – AnyAD
    Dec 12 '18 at 3:48
















1












1








1





$begingroup$


Is it possible? I know that for a matrix to be diagonalizable it needs sufficient linearly independent eigenvectors to make up invertible matrix P, but does that mean B itself must also be invertible?






linear-algebra diagonalization







share|cite|improve this question









$endgroup$




Is it possible? I know that for a matrix to be diagonalizable it needs sufficient linearly independent eigenvectors to make up invertible matrix P, but does that mean B itself must also be invertible?






linear-algebra diagonalization




linear-algebra diagonalization






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asked Dec 12 '18 at 3:45









SolidSnackDriveSolidSnackDrive

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  • 1




    $begingroup$
    Yes it is. For example take the matrix $[1 0; 0 0]$.
    $endgroup$
    – AnyAD
    Dec 12 '18 at 3:48
















  • 1




    $begingroup$
    Yes it is. For example take the matrix $[1 0; 0 0]$.
    $endgroup$
    – AnyAD
    Dec 12 '18 at 3:48










1




1




$begingroup$
Yes it is. For example take the matrix $[1 0; 0 0]$.
$endgroup$
– AnyAD
Dec 12 '18 at 3:48






$begingroup$
Yes it is. For example take the matrix $[1 0; 0 0]$.
$endgroup$
– AnyAD
Dec 12 '18 at 3:48












3 Answers
3






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5












$begingroup$

A matrix $A$ is said to be diagonalizable, if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$.



Note that $A$ is invertible if and only if $D$ is invertible. Therefore, the question boils down to : are all diagonal matrices invertible?



The answer is no, simply because a diagonal matrix is invertible if and only if every diagonal element is invertible. But if some diagonal elements are zero, then the diagonal matrix will not be invertible. Therefore, neither will $A$.



For example, consider the $2 times 2$ matrix $E_{11}$, having a $1$ at the position $11$ and zero elsewhere. Then, $E_{11}$ is a diagonal non-invertible matrix, as the diagonal entry $(E_{11})_{22} = 0$.



Now, $A = D$, or $A=PDP^{-1}$ for any invertible $P$ serves as a counterexample.



Also, note that the zero matrix is diagonalizable, in fact diagonal, but not invertible. This would be the most obvious example.






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    $begingroup$

    Let $$B= begin{bmatrix} 0 & 0 \ 0 & 0end{bmatrix}.$$



    It is a diagonal matrix and it is diagonalizable, we can let $P$ be any non-singular matrices, but it is not invertible.






    share|cite|improve this answer









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      1












      $begingroup$

      Yes it is possible. Say we have a diagonal matrix $D=mathrm {diag}(2,0)$, then $D$ is not invertible. Let $P$ be some invertible matrix, say
      $$
      P = begin{bmatrix}
      a & b \ -b & a
      end{bmatrix}
      $$       where $a^2 +b^2 neq 0$, then $A = PDP^{-1}$ works.






      share|cite|improve this answer









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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        5












        $begingroup$

        A matrix $A$ is said to be diagonalizable, if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$.



        Note that $A$ is invertible if and only if $D$ is invertible. Therefore, the question boils down to : are all diagonal matrices invertible?



        The answer is no, simply because a diagonal matrix is invertible if and only if every diagonal element is invertible. But if some diagonal elements are zero, then the diagonal matrix will not be invertible. Therefore, neither will $A$.



        For example, consider the $2 times 2$ matrix $E_{11}$, having a $1$ at the position $11$ and zero elsewhere. Then, $E_{11}$ is a diagonal non-invertible matrix, as the diagonal entry $(E_{11})_{22} = 0$.



        Now, $A = D$, or $A=PDP^{-1}$ for any invertible $P$ serves as a counterexample.



        Also, note that the zero matrix is diagonalizable, in fact diagonal, but not invertible. This would be the most obvious example.






        share|cite|improve this answer









        $endgroup$


















          5












          $begingroup$

          A matrix $A$ is said to be diagonalizable, if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$.



          Note that $A$ is invertible if and only if $D$ is invertible. Therefore, the question boils down to : are all diagonal matrices invertible?



          The answer is no, simply because a diagonal matrix is invertible if and only if every diagonal element is invertible. But if some diagonal elements are zero, then the diagonal matrix will not be invertible. Therefore, neither will $A$.



          For example, consider the $2 times 2$ matrix $E_{11}$, having a $1$ at the position $11$ and zero elsewhere. Then, $E_{11}$ is a diagonal non-invertible matrix, as the diagonal entry $(E_{11})_{22} = 0$.



          Now, $A = D$, or $A=PDP^{-1}$ for any invertible $P$ serves as a counterexample.



          Also, note that the zero matrix is diagonalizable, in fact diagonal, but not invertible. This would be the most obvious example.






          share|cite|improve this answer









          $endgroup$
















            5












            5








            5





            $begingroup$

            A matrix $A$ is said to be diagonalizable, if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$.



            Note that $A$ is invertible if and only if $D$ is invertible. Therefore, the question boils down to : are all diagonal matrices invertible?



            The answer is no, simply because a diagonal matrix is invertible if and only if every diagonal element is invertible. But if some diagonal elements are zero, then the diagonal matrix will not be invertible. Therefore, neither will $A$.



            For example, consider the $2 times 2$ matrix $E_{11}$, having a $1$ at the position $11$ and zero elsewhere. Then, $E_{11}$ is a diagonal non-invertible matrix, as the diagonal entry $(E_{11})_{22} = 0$.



            Now, $A = D$, or $A=PDP^{-1}$ for any invertible $P$ serves as a counterexample.



            Also, note that the zero matrix is diagonalizable, in fact diagonal, but not invertible. This would be the most obvious example.






            share|cite|improve this answer









            $endgroup$



            A matrix $A$ is said to be diagonalizable, if there exists a diagonal matrix $D$ and an invertible matrix $P$ such that $A = PDP^{-1}$.



            Note that $A$ is invertible if and only if $D$ is invertible. Therefore, the question boils down to : are all diagonal matrices invertible?



            The answer is no, simply because a diagonal matrix is invertible if and only if every diagonal element is invertible. But if some diagonal elements are zero, then the diagonal matrix will not be invertible. Therefore, neither will $A$.



            For example, consider the $2 times 2$ matrix $E_{11}$, having a $1$ at the position $11$ and zero elsewhere. Then, $E_{11}$ is a diagonal non-invertible matrix, as the diagonal entry $(E_{11})_{22} = 0$.



            Now, $A = D$, or $A=PDP^{-1}$ for any invertible $P$ serves as a counterexample.



            Also, note that the zero matrix is diagonalizable, in fact diagonal, but not invertible. This would be the most obvious example.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Dec 12 '18 at 3:59









            астон вілла олоф мэллбэргастон вілла олоф мэллбэрг

            38.3k33376




            38.3k33376























                2












                $begingroup$

                Let $$B= begin{bmatrix} 0 & 0 \ 0 & 0end{bmatrix}.$$



                It is a diagonal matrix and it is diagonalizable, we can let $P$ be any non-singular matrices, but it is not invertible.






                share|cite|improve this answer









                $endgroup$


















                  2












                  $begingroup$

                  Let $$B= begin{bmatrix} 0 & 0 \ 0 & 0end{bmatrix}.$$



                  It is a diagonal matrix and it is diagonalizable, we can let $P$ be any non-singular matrices, but it is not invertible.






                  share|cite|improve this answer









                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    Let $$B= begin{bmatrix} 0 & 0 \ 0 & 0end{bmatrix}.$$



                    It is a diagonal matrix and it is diagonalizable, we can let $P$ be any non-singular matrices, but it is not invertible.






                    share|cite|improve this answer









                    $endgroup$



                    Let $$B= begin{bmatrix} 0 & 0 \ 0 & 0end{bmatrix}.$$



                    It is a diagonal matrix and it is diagonalizable, we can let $P$ be any non-singular matrices, but it is not invertible.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Dec 12 '18 at 3:57









                    Siong Thye GohSiong Thye Goh

                    101k1466118




                    101k1466118























                        1












                        $begingroup$

                        Yes it is possible. Say we have a diagonal matrix $D=mathrm {diag}(2,0)$, then $D$ is not invertible. Let $P$ be some invertible matrix, say
                        $$
                        P = begin{bmatrix}
                        a & b \ -b & a
                        end{bmatrix}
                        $$                         where $a^2 +b^2 neq 0$, then $A = PDP^{-1}$ works.






                        share|cite|improve this answer









                        $endgroup$


















                          1












                          $begingroup$

                          Yes it is possible. Say we have a diagonal matrix $D=mathrm {diag}(2,0)$, then $D$ is not invertible. Let $P$ be some invertible matrix, say
                          $$
                          P = begin{bmatrix}
                          a & b \ -b & a
                          end{bmatrix}
                          $$                           where $a^2 +b^2 neq 0$, then $A = PDP^{-1}$ works.






                          share|cite|improve this answer









                          $endgroup$
















                            1












                            1








                            1





                            $begingroup$

                            Yes it is possible. Say we have a diagonal matrix $D=mathrm {diag}(2,0)$, then $D$ is not invertible. Let $P$ be some invertible matrix, say
                            $$
                            P = begin{bmatrix}
                            a & b \ -b & a
                            end{bmatrix}
                            $$                             where $a^2 +b^2 neq 0$, then $A = PDP^{-1}$ works.






                            share|cite|improve this answer









                            $endgroup$



                            Yes it is possible. Say we have a diagonal matrix $D=mathrm {diag}(2,0)$, then $D$ is not invertible. Let $P$ be some invertible matrix, say
                            $$
                            P = begin{bmatrix}
                            a & b \ -b & a
                            end{bmatrix}
                            $$                             where $a^2 +b^2 neq 0$, then $A = PDP^{-1}$ works.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Dec 12 '18 at 3:50









                            xbhxbh

                            6,1251522




                            6,1251522






























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